Introduction to Matrix Algebra

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STUMNTS TEXT

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INTRODUCTIO;; TO MATRIX ALGEBRA

SCHOOL MATHE#ATlCS STUDY

YALE UNIWRSCTY ?RESS

GROUP

School Mathematics Study Group

Introduction to Matrix Algebra

Ii~troductionto Matrix Algebra Strrdetlt's

Text

Prcparcd u n h r th ~est~pen.kjon of rhc Panel on Sample T e x r h k s

of rhc School Mathcmarm Srudy Group: Frank

B. Allen

Edwirl C. Douglas Donakl E. Richmond Cllarlm E,Rickart Hcnry Swain Robert J. Walkcr

Lyonr Township High

School

Tafi S C h I Williamr Collcgc Yale Univcrriry New Trier Township High S c h d Cornell University

Ncw Havcn a d h d o n , Y a1c University Press

Coyyrig!lt O I*, r p 5 r by Yak Ut~jversiry. Printed in rhc Ur~itcdStarm af America. A]] r i ~ h t srescwcd. Ths h k may not ltrc repcduccd, in whdc or in part, in

any form, wirho~~t written permission from the publishen. FinanciaZ slspporr for the S c b l Mathemaria Study Group has km providcd by rhe Narional Scicncu: Foundation.

The i n c r e a s i n g contribution of mathematics to the culture of the d c r n w r l d , as well as i t s importance as a v i t a l part of s c i e n t i f i c and humanistic tdueation, has mndc i t essential that the m a t h e ~ u t i e ai n our echool@ be bath wZ1 selected and e l l taught. t j i t h t h i s i n mind, the various mathc~atical organLtaeions i n the United Seatss cooperated i n the forumtion of the School F ~ t k m a e i c sStudy Group (SPSC). SMG includes college and unirersiley mathmaticlans, eencbera of m a t h w t i e s a t

a l l lavels, experts i n education, and reprcscntarives of science and tcchrmlogy. Pha general objective o f SMG i a the i m p r o v m n t a f thc? teaching of =themtics in the sehmla of t h i s country. The l a t i o n u l Science Foundation has provided subs tantiaL funds for the support of this endeavor. One of the pterequlsitea for the improvement of t h e teaching of math&matira in our ~ c h m t si e on improved c u r r i e u ~ ~ n which s takea account of thc incrcasing use of oathematice Ln science and teehnoLogy end i n other areas o f knowledge and a t the a m timc one v f i i ~ href l o e t a recent advancta i n mathem~tica i t s c l f . One of thc f l r n t projects undartaken by SrSC was t o enlist a group of outstanding mathematiclans and nrarhematica teachern t o prepare a nariea of textbooks uhich would illustrmre such A n improved curriculum.

The professional arathcmatLcLanlr in SMSG belleve that the mathematics p r P eented In this t t x r is vaLuahLc fat a l l w2 l-ducatcd citlzene i n our society t o know 6nd t h a t it ia tqmrtant f o r the prceollage atudent: to learn i n preparation far advanced vork i n thc f i e l d . A t thc time t i n e , teachers in SWC believe t h a t it is presented in rrueh a farm that i t can ha r ~ a d F l ygrasped by students.

! !

In most inntanees the mnterlal will have a inmiltar n o t e , he the prcsentation and the point of vicv vill be diftcrsnt. Samc matcrial w i l l be e n t i r e l y neu t o ehc traditional curriculum, This i s ar i t should be, for ~ A a t h c m & t i ~i & a a ltving and an cvcwrawinff subject, and Rat a dead and frozen product of ant i q u i t y . This heakthy fusion o f the old and Ehc ncu should lead students to a better undcsstsnding of the basic concepts and structure of mthemtics and provide a f i m r foundation for understanding and use of ~ t h c m a t i c sin a gcitntlfic sociery.

It la n o t intendad that this book be regarded an the mly d c f i n i t l v a way of presenting good mathcmatica t o student# a t t h i s Level. Instead, I t should be thought of as e ample of the kind of i q r w e d currieulua that we need and as a source of nuggestiana for the authora of c-rclal textbka. It is sincerely hoped that these t e x t s wLl1 lead the way coward inspiring a more meaningful reaching o f Hathemnckea, t h e Wcca and Servant of tho Sctencea.

5.

(b)

Name the entries in the 3rd row.

(c)

Name the entries in the 3rd column.

(d)

Name the entry

(e)

Forwhatvalues

i, j

is

b

(f)

For what values

i, j

is

b

(g)

W r i t e the transpose

(a)

Write a

3 x 3

matrix a l l of whose entries are whole numbers.

(b)

Write a

3

4

matrix none of whose entries are whole numbers.

(c)

Write a

5 x 5

matrix having a l l entries in i t s f i r s t two rows

X

b12. ij

ij

+0?

= O?

B ~ ,

p o s i t i v e , and a l l entries in its l a s t three rows negative. 6.

1-3.

(a)

How many entries are there in a

(b)

In a

(c)

Xn en

(d)

In an rn x n matrix?

4 x 3 n x n

2 x 2

matrix?

matrix? matrix?

Equality of Matrices

Two matrices are equal provided they are

of

the same order and each entry For example,

in the f i r s t is equal t o the corresponding entry Ln the second.

but

Definition 1-2.

Two matrices

A

and

B

are equal,

A =

B,

if and only

if they are of the same order and their corresponding entries are equal.

Thus,

36 ordinary algebra of numbers insofar as multiplication is concerned? L e t us consider an example that will y i e l d an answer t o the foregoing

question.

Let

If we compute

AB,

we find

Now, if we reverse the order of the factors and compute

AB

Thus

BA,

we find

BA are d i f f e r e n t matrices!

and

For another example, let

I

Then

while

Again

AB and BA are d i f f e r e n t matrices; they are not even of the same order! Thus we have a first difference between matrix algebra and ordinary

algebra, and a very significant difference it is indeed.

men

we

numbers, we can rearrange factors s i n c e the commutative law h o l d s : x

E

R

and

y

E

R,

we have

xy = yx.

multiply real For all

When we multiply matrices, w have no

37 such law and w e must consequently b e careful t o take the factors i n the order given.

W e must consequently d i s t i n g u f s h between the reaul t o f multiplying

on the right by by

A

to g e t

A

to g e t

and the r e s u l t of multiplying

BA,

B

B

on the left:

In the algebra of numbers, these two operations of "right

AB.

m u l t i p l i c a t i o n t 1 and "left muLtiplFcation" are the same; in matrix algebra, they

are n o t necessarily the same. Let u s explore some mote differences!

A =

Patently,

[; ;]

# 0 and B # g.

A

thus, we f i n d

and

8 =

Let

[-;_93]

But i f we compute

AB,

. we obtain

. Again, l e t

d

Then

The second major difference between ordinary algebra and matrix algebra is that the product of tw, matrices can be a zero matrix without either factor

being a zero matrix. The breakdown f o r matrix algebra o f the law that

that

xy = 0

only i f either

x

or

y

xy = y x

is zero causes additional difference^.

For instance, f o r real numbers ue know that if ab = a c , then

b =

c.

Proof,

and

a

+ 0,

T h i s property is called the cancellation law for multiplication.

We d i v i d e d the proof into afmple atepB:

(b)

ab ac, ab - a t = 0,

(c)

a(b

{a)

and of the law

- c)

= 0,

For tmtricer , the above s t t p from (el t a Cd) f a i l d and rhs proof i e not v a l i d . In fact, AB can be aqua1 t o AC, with A 3 2, y e t B C g Thud, l e t

+

Then

but

Let us consider another d i f faranee. have a t moer

equation Praof

mt

.

t

v

Wi h o w t h t r real

nmber

a

can

that is, thars arc a t m o s t tw rocrre of the

square ~ root.;

a.

Again, ws give the r i q l h s t s p l o f the proof: Supp6c that

py

-

them

a;

W,

XX

xx-yy-0, (X

Yx

-

-

Y)(X

+ Y ) ' = + (v+ x y )

KY.

Prm (dl and

la),

Prom (cl and ( I ) , Therefore, sither

- y)(x

(x

(x

+ y)

- y)(x + y)

- yy,

-+

- --

x

-y

Therefore, either x

y

0

or

or x

m0.

n.

y

0.

x

m

y.

For matrices, statement (c) is f a l s e , and therefore the steps t o ( f ) (g) arc i n v a l i d .

Even i f ($1 vtre valid, the #tap from (g) t o (h) f a i l & .

ISM. 1-81

and

39 Thsrefore, the forcgofng proof i e invalid if ue t r y t o a p p l y i t t o m t r i c a a . fact, k t is false that a matrix can have a t =st t w o square roots: b k have

In

Ttluo t h o matrix

has thc four d i f f e r a n t square roots

There are mra!

By giving

x

Given any number

x

#

0,

we

have

ady one of an i n f i n i t y of different real values, w obtain an

i n f i n i t y o f d i f f e r e n t square roots a f the matrix 1:

Thub

roots!

the very stnple

You can see,

most t w o squaro

2 x 2 matrix

I has i n f i n f t e l y many d i s t i n c t square

then, that the fact that a real or c q l c x nunbct has a t

m a t e Lsr by no means trivial.

2.

k k c the cakculations of EKrrcts-a L for tha artrictr

3.

k t

and

A

Calculate

B be as in Excrciae

AI,

2, and let

IB, and

IA, RX,

(AX)B.

4. Ltt

Show by c m p u t a t i o a t h a t

where

(A)

(A

+ R I M + BS

+ A'

(b)

(A

+ B)(A-

+ -

A

and

02

B)

denole

+~

M

A B+ 8'.

and

BB, rempectlvtly.

6.

Find a t Least t3 squara roots o f the m t r i x

7.

Show that: the m t r i x

a a t i s f l c s the aquatian A'

-

2.

How many 2 x 2 matrices .atla

fying

thia

equation can you flad?

8.

Show t h a t the matrix

s a t i a f i e r the cquarion k3

1-9.

0.

Properties of Hatrix Hultiplication (Concluded) tb have seen that tvo basic laus governing multiplication i n the algebra

of ordinary ambers break dowa when k t comas t o matrices.

and the cancellation law do n o t hold.

t h e c-tativc

law

A t t h i ~point, you night fear a total

all thc other f m i l i a r law.

This Is not the c a m . Aside from the tw laws aentloned, and the f a c t that, as wc ehall rec later, many matrices do not have aulsiplicativc invarues (reciprocals), thc other bani c Paw o f ardinary algebrn generally r-in v a l i d for mutriccs. The assoedative Law holda for the w l t i p l i e a t i m of mtricea and thare are dtatributive levr that u n i t e addition and multiplication. A feu e x m p l r s will aid us In understanding the lam. collapas

of

kt

[see.

1-81

42 Then

and

Thus, in this case,

and

so that also

Since multiplication is not c m r a t i v e , we cannot conclude from Equation

(2)

that the distrtbutive principle is v a l i d with the factor

hand s i d e of

B

+ C.

A

on the right-

Having illustrated the Left-hand distributive law, w

43 now i l l u s t r a t e the right-hand distributive law with the following example: We have

and

Thus, (I3

+ C)A

= BA

+ CA.

You might note, in passing, t h a t , i n the above example,

These properttes o f matrix mu1tiplication can be expressed as theorems, as f o l l o w s .

Theorem 1-55.

If

then

(AB)C

Proof. (Optional .) We have

r:

A(BC)

.

Since the o r d e r of a d d i t i o n is arbitrary, we know that

Hence,

Theorem 1-6.

then

A ( B f C) = AB

Proof.

If

+ BC.

(Optional.)

(8

We have

+ C)

=

[bjk

+

'jk]

p x n'

aij ( b j k + c . ) J~

1

mxn

] ...

-

AB

f AC.

Hence,

Theorem 1-7.

If

= then

(B

+ CIA

[bjk] p x n *

= BA

=

['jk]

p ~ n 'and

A =

[%i]

nxq

+ CA.

Proof.

The p r o o f is similar t o that o f Theorem 1-6 and will be l e f t as an exercise for the s t u d e n t .

It should be noted t h a t if the

comtative

Law h e l d for matrices, it would

be unnecessary to prove Theorems 1-6 and 1-7 separately, since the two stare-

men ts

would be equivalent.

For matrices, however, the two statements are n o t e q u i v e

l e n t , even though borh are true. statements like

and

can be false for matrices.

The order of factors is most important, since

46 m x n

Earlier we defined the zero matrix o f order

and showed that it: i s

the identiry element for matrix addition:

where A i s any matrix of order m x n. This zero matrix plays the same role in t h e m l t i p l f c a t i o n of =trices as the number zero does in the ~mltiplicatition of real numbers. For exrtm~le,we have

Theorem 1 4 .

For any matrix

we have

Omxp

The proof is

A p x n = 0m x n and A p x n 0n x q = 0p x q '

easy and is l e f t t o the student.

Now we may be wondering If there is an i d e n t i v element for the multiplica-

tion of matrices, namely a matrix that p l a y s the same r o l e a e the number 1 does

in the multiplication of real numbers.

There

(Far all real numbers

a, l a = a = al.)

such a matrix, c a l l e d the unit matrix, or the identity matrix for

multiplicatfon, and denoted by the symbol I.

The matrix

12, namely,

is called the u n i t matrix of order 2 .

The matrix

is called the unit matrix o f order 3 .

In general, the unit marrix o f order

n

i s the aquare matrix

[eij] nx n

such that

e

ij

1

forall

i - j

and

e

ij

forall

= Q

#

j (

i

2

.

n

; =

1

,

.n

W e n o w s t a t e the

important property o f the unit matrix as a theorem. Theorem 1-9.

Proof.

product

all

Am

k

j

If A

is an m x n

matrix, then

By d e f i n i t i o n , t h e e n t r y in the i s the sum

i-th

AIn = A

row and

j-th

and

ImA

A.

column of the

+ a e + . . . + a e . Since e ilelj 12 2j i n nj' kj = 0 a l l terms but one in this Bum are zero and drop o u t . We are a

for

Thus the entry in the i-th row and j-th column of the a .e = a $J j j ij' product is the same as the corresponding entry in A. Hence A l n = A . The leftwith

equality

Imh = A

may be proved the same way.

In most situations, it: is

not

necessary t o specify the order of the unit matrix since the order is inferred from the context.

Thus, for ImA = A

AI,,

we write

IA * A = AX. For example, we have

and

Exercises 1-9

43 Test the formulas

+ C)

A(B

+ AC,

= AB

(B 4- CIA = BA

+ CA,

A(B 4- C)

AB

+ CA,

= BA

+ CA.

-

+ C)

A(O

Which are correct, and which are false? 2.

Let

A Show that 3.

2,

AB

001

[; but

BA =

Show t h a t f o r all matrices

A

[;

and

;I.

.==

2. of the form

and

B

AB =

BA.

we have

Illustrate by assigning numerical values to a, a, b, c, and d i n t e g e r s . 4.

Find the value of x

5.

For the matrices

show rhat 6.

b,

and

c,

for which the following product is

d,

1:

0

0

0

0

0

0

0

0

-

0

1

0

0

1 2 0

=

CA,

AB

1

BA,

rhat

AC

and that

Show t h a t the matrix

fsec. 1-93

BC

1

CB.

0

0

with

= I.

satisfies the equation

Find at least one more s o l u r i o n of t h i s

equation.

7.

Show t h a t for all matrices

A

of the form

we have

Illustrare by assigning numerical values to 8.

a

and

b.

Let

Compote the following:

If

AB =

- BA,

A

ED,

(el

m,

( f

FE.

B are s a i d

and

c l u s i o n s can be drawn concerning

'I. Show that the m a t r i x A = A'

(a>

- 5A + 71 = 2.

D,

[-: :]

.-

t o be

bhat: con-

and

E,

F?

is a solution o f the equation

10. Explain why, in matrix algebra, (h 4- 8) ( A - B )

except in special cases.

# A

2

- B

2

Can you devise two m a t r i c e s

A

and B

will i l l u s t r a t e the inequality? Can you d e v i s e two matrices

A

that and

B

50 that will illustrate the special case?

(Hint:

Use square metrices of

order 2.) *

11.

Show that if V and W

12,

Prove that

are n x 1

colum vectors, then

( A B ) ~= B ~ A ~asrruming , that

A

and B are c c m f o l l e for

multiplication,

no tation, prove the right-hand d i s t r i b u t i v e law (Theorem 1.7)

13. Using

Sumwry

1-10.

In this introductory chapter

ve have d e f i n e d several operations on

matricea, such as a d d i t i o n and multiplication.

Theee operatione d i f f e r from

those of elementary algebra i n tbat they cannot always be performed. we

.

do not add a

matrix and a 4 x 3

nor

2 x 2

3 x 4

3 x 4.

matrix t o a

3 x 3 matrix; again, though a

Thus,

4x 3

matrix can be multiplied together, the product i e neither More importantly, the c-tative law for multiplication

do not hold. There La a third significant difference that we shall explore more f u l l y

and the cancellation law

in later chapeere but shall tntroduce now. Recall that the operation of subtraction was closely associated with that o f addition. In order t o solve equations of the form

it fsconveniept t o employ the additive inveree, or negative,

4.Thus, if

the foregoing equation holds, then we have A+x+(-A)

X + A + (-A)

x+g=

-

B+(-A),

= B+(-A),

B-A*

X-I-A. b you know, every matrix A

e negative -A*

Now "division" i s closely

9 associated w i t h m l t i p l i c a t i o n in a parallel manner.

In order to solve

equation8 o f the form

multiplicative inverse (or reciprocal), which is

we would analogously employ

denoted by the symbol A-'.

The defining property is A-'

A = I = M - ~ . This

enables us to solve equations of the form

Thus i f the foregoing equation holds, and i f A has a mu1 tiplicarive Lnverse -1 A , then

Now, many matrices other than the zero matrix

2

do not possess multiplicative

inverses; for i n s mnce,

are matrieee of this sort.

This fact constimtes a very significant difference

between the algebra o f -trices

and the algebra of real numberr.

In the next

tvo chapters, we shall explore the problem of matrix inversion in depth.

Before closing this chapter, w e should note that matrices arising in s c i e n t i f i c and industrial applications are narch larger and their entries much

more complicated than has been the case in this chapter.

As you can imagine,

the computations involved d e n dealing with larger matrices (of order 10 or more), which is usual in applied work, are ao extensive a8 to discourage their uea i n hand computations.

Fortunately, the recent development of higltapeed

electronic computers has largely overcome this difficulty and thereby has made it mre feasible t o apply matrix methods i n many areas of endeavor.

Chapter 2

THE ALGEBRA OF 2-1

.

2

X

2 MTRICES

Introduction

In Chapter 1,

we considered the elementary operations of addition and

This algebra is similar in many respecta to the algebra of real numbers, a 1 though there are important differ-ences. Specifically, we noted that the connnutative law and the cancellation

multiplication for rectangular matrices.

law do not hold f o r matrix multiplication, and that division is not always possible.

With matrices, the whole problem of d i v i s i o n i s a very complex one; i t i s

centered around the existence of a multiplicative inverse. question:

Let us ask a

If you were given t h e matrix equation

could you solve i t for the unknom

4 x 4 matrix

X?

Do n o t be dismayed

Eventually, we s h a l l learn methods of solving t h i s

if your answer is "No."

equation, but the problem is complex and Lengthy.

In order to understand

this problem in depth and at the same time comprehend the f u l l significance

of the algebra we have developed so far, we shall largely confine our attention i n t h i s chapter t o

a special subset of the set of all rectangular matrices; 2 x 2

namely, we shall consider the a e r of

square matrices.

When one s tsnds back and takes a broad view of the many d i f f e r e n t kinds of numbers that have been studied, one sees recurring patterns. us look at the rational numbers for a moment.

can add and m l t i p l y . granted.

This

statement

But i t is not true of a l l

For Lnstance, let

Here is a set of numbers that

WE

is so simple that we almost rake it for

s e t s , so let:

us g i v e a name to the n o t i o n

that: is involved.

Definition 2 4 . a f i r s t member

(i)

a

of

A set S

S

i s said to be closed under an operation

and a second member b

the operation can be performed on each

of a

S

if

and b

of

S,

R on

5rrf a r each

(ii) member of

a

and

b

of

the r e s u l t of the operation i e a

S,

S.

For example, the set o f p o s i t i v e i n t e g e r s i s not closed under the operation

of division since for some p o s i t i v e integers

a

and

b

the ratio a/b

i s not

a positive integer; neither is the s e t of rational nmbers closed under division, since the operation cannot be perfomed if

b = 0 ; but the set of positive

rational numbers is closed under d i v i s i o n since the quotient o f two positive

rational numbers is a positive rational number. Under addition and multiplication, the s e t of rational numbers s a t i s f i e s the following postulates:

The s e t is closed under addition. Addition i s c o m t a t i v e . Addition i s a s a o c i a r i v e .

There i s an i d e n t i t y member ( 0 ) for addition.

There is an additive inverse member

-a

for each member

a.

The set i s closed under multiplication. Mu1 t i p l i c a t i o n is cwnattative

.

Efultiplication i s a s s o c i a t i v e .

There is an i d e n t i t y member (1) for multiplication. -1

There i s a multiplicative inverse member a other than 0 .

for each member

a,

. . . Multiplication is d i s t r i b u t i v e over a d d i t i o n . Since there exists a rational multiplicative inverse for each rational number except

0,

division (except by 0) is always possible in the algebra of

rational numbers.

where

a

and

b

In other words, all equations of the form

are rational numbers and

the algebra of rational numbers.

a # 0, can be solved for

For example:

x

in

I n order t o solve the equation

we multiply both sidea of the equation by

of - 2 1 3 .

- 3/2,

the m u l t i p l i c a t i v e inverse

Thus we obtain

which i s a rational number. The foregoing set of postulates is satisfied also by the s e t of real numbers.

Any s e t that satisfies such a e e t of postulates i s called a f i e l d .

Both the s e t o f real numbers and the

s e t of r a t i o n a l s , which i s a subset

the s e t of real numbers, are fields under addition and m u l t i p l i c a t i o n .

are many systems t h a t have this same pattern.

of There

In each of these systems,

division (except by 0) is always possible. Now our imediate concern is to explore the problem of division in the set

of matrices.

There is no blanker answer that can readily be reached, although

there i s an answer that we can f i n d by proceeding stepwise. limit our discussion to the s e t of

2 x 2

matrices.

A t f i r a t , l e t us

We do t h i s n o t o n l y to

consider division in a smaller domain, but a l s o to study i n d e t a i l the algebra associated w i t h t h i s subset.

A more general problem o f . m t r i x d i v i s i o n will be

considered in Chapter 3 .

Exercises 2-1 1.

Determine which of the following sets are closed under the s t a t e d operation: (a)

the s e t of integers under addition,

(b)

the s e t of even numbere under multiplication,

(c)

the s e t 111 under multiplicarion,

(d)

the s e t of positive irrational numbers under division,

[ssc. 2-11

56

2.

(e)

the s e t o f integers under the operation of squaring,

(f)

the s e t of numbers A = Ex: x > 3)

under addition.

Detennine which of the following statements are true, and s t a t e which of

the indicated operations are c o m u t a t t v e : (a)

2-3=3-2,

(b)

4+2=2+4,

(c)

3+2=2+3,

(d)

x a

(e)

a

(f)

Pq = q p ,

-b

Jb

=

= b

- a,

+ Ja, a

a

and

and

b

positive,

real,

b

real,

P and q

F le 2 = 2

(g)

3.

+

+ fi.

Determine which of the following operations 3, defined for p o s i t i v e integers in terns of a d d i t i o n and mu1tlplication, are c m u t a t i v e : (forexample,

(a) x 3 y P x + 2 y

4.

(e)

x * y = x Y,

(f)

x * y = x + y + l .

Determine which of the following operations

233=2+6=8),

*,

d e f i n e d for p o s i t i v e

integers i n terms o f addition and mu1 tipLfcation, are associative:

5.

(a)

x * y = x + Z y

(d)

x

(el

x

(f)

x

*

Y

*Y *y

(for enample, ( 2

*

3)

*

4 = 8

*

4

a

161,

XI

=

J G s

= xy

+ 1.

Determine whether the operation

*

9, that is, determine whether

*

x

is distributive over the operation (y 5

z) = ( x

* y) + ( x *

2)

and

*

( y f z)

x = (y

* x)

%

( Z

*

X)

,

fi.

*

and

where the operations

are

d e f i n e d For positive integers i n terms o f addition and mu1 tiplicetion of real numbers: x +

(a)

x + y =

(b)

x3yE2x+2y,

(c)

x * y =

Why is the answer

x +

x*yEllIY;

y,

x

y+1,

*

y

=

1

xy;

x * y - x y .

the same I n each case for Left-hand d i s t r i b u t i o n as it

is for right-hand d i s rribution? 6.

In each of the following examples, determine if the specified s e t , under a d d i t i o n and multiplication, constitutes a f i e l d : (a)

the s e t of all positive numbers,

(b)

t h e set: of all rational numbers,

(c)

the set of ell reel numbers of the form a and b are integers,

+ bfi,

the e e t of all complex numbers o f the form a and b are real numbers and i =

(d)

a

2-2.

a

The Ring o f

2 x 2

Matrices

Since we are confining our attention t o the subset of

it i s very convenient t o have a symbol for this subset, s e t of all

2 x 2

matrices.

If

A

f

where

bi,

2 x 2

tle let

where

matrices,

M

denote the

is a member, or element, of t h f s s e t , we

express t h i s membership symbolically by

A E M.

S i n c e all elements of

M

are

matrices, our g e n e r a l definitions of addition and multiplication hold for t h i s subset.

The s e t

M

is not a f i e l d , as defined in Section 2-2,

since M

does n o t

have a l l the properties of a f i e l d ; for example, you saw in Chapter 1 t h a t

multiplication is not commutative in

we have

M.

Thue, for

Let us now consider a less r e s t r i c t i v e sort of mathematical system known as a ring; - t h i s name is

Definition 2-2.

usually aktribured t o David Hilbert (1862-1943).

A r i n g is a s e t with two operations, called addition and

mu1tiplication, that possesses the following properties under a d d i t i o n and

multiplication: The s e t is closed under a d d i t i o n .

Addition is c o m t a t i v e

.

Addition i s associative

.

There is an i d e n t i t y element for addition. There is an additive inverse for each element.

The s e t is closed under mu1 tiplieation. Multiplication is associative.

Multiplication is distributive over addition. Does the s e t

M

s a t i s f y these properties?

Consider the s e t of a l l real numbers.

b u t the answer i a not q u i t e obvious. '

It seems clear t h a t i t does,

This s e t is a field because there e x i e t s , among other things, an additlve inverse for each number in thfs set. o f the real numbers.

Now the p o s i t i v e integers are a subset

Does thLs s u b s e t contain an a d d i t i v e inverse for each

element? Since we do not have negative integers i n the s e t under consideration, the answr i e "No"; therefore, the s e t of positive integers irr not a f i e l d . Thus a subset does not: necessarily have the same propertlea aa the complete set.

To b e certain that the s e t

M

is a ring, ue must systematically make sure

that each ring criterion is s a t i s f i e d .

For the m a s t part, our proof w i l l be a

reiteration of the material i n Chapter 1, s i n c e the general properties o f matrices w i l l be valid for the subset 2 x 2

matrices i s a

2 x 2

M

of

2 x 2

matrices.

matrix; that i s , the set

For ewmple, [aec. 2-21

i6

The sum of twu

c l o s e d under a d d i t i o n .

fie general proofs of cormartativity and associativity are valid.

The unit

matrix is

the zero matrix i s

and the additive inverse of the matrix

When we consider the multiplication of

2 x 2

matrices, we must first v e r i f y

that the product is an element of this s e t , namely a

2 x 2

matrix.

&call

that the number of rows in the product is equal t o the number of rows in the

lef t d n d factor, and the number o f coluarms i a equal t o the number of columns in the right-hand factor. Thus, the product of two elements of the s e t M must be an element of this s e t , namely a

is c l o s e d under mu1tiplication.

2 x 2

matrix; accordingly, the s e t

For example,

The general proof of a s s o c i a t i v i t y i s valid for elements of

for rectangular matrices.

of M

M,

s i n c e it , i s v a l i d

Also, both of the distributive laws hold for elements

by the same reasoning.

For example, t o illustrate the ascrociative law

for multiplication, ue have [see. 2-21

and also

and t o illustrate the left-hand distributive Law, we have

and also

Since we have checked that each of the ring postulates i s fulfilled, we have shown that the s e t

multiplication. Theorem

M of

2 x 2

matrices is a ring under addition and

We s t a t e this result fonaally as a theorem.

2-1.

The set M

2 x 2

of

matrices i s a ring under addition

and trmltiplication. Since the list o f defining properties for a f i e l d contains a l l the d e f i n i n g propertier for a ring, i t follows that every f i e l d is a r i n g .

statement i a n o t true; for example, we now know that the s e t matrices i s a ring but not a field.

that is, for each

U = Thus the s e t

M is

a ring

M

of

2 x 2

The set H bas one more of the f i e l d

properties, namely there is an i d e n t i t y element

for multiplication in M;

But the converse

A

-

A

€

M we have

AX.

with an identity element. [see.

2-21

61 A t this time, we should verify that the comnutative law for multiplication

and the cancellation law are not valid in H by giving counterexamples.

Thus

we have

but

so that the commutative Law for multiplication does n o t hold.

Also,

so that the cancellation law does n o t hold;

Exercises 2-2

1.

Determine i f the set of a l l integers

18

a ring under rhe operations o f

addition and multiplication. 2.

Determine which of the fallowing s e t s are rings under a d d i t i o n and

mu1 tiplication:

3.

the form a

+b

A,

where

(a)

the s e t of numbere are integers ;

(b)

the s e t of f o u r f o u r t h roots of unity, namely, and -i;

(c)

the s e t of numbers

of

a/2,

where

a

e

and

+L, -1,

b i,

is an integer.

Determine If the set o f a11 u t r t c e ~o f the fom

[t :],

with

a e R,

forms a ring under addition and multiplication as defined for matricea.

4.

Determine i f the s e t of a l l matricea of the form

forms a ring under addition and multiplication a@ defined for matrices.

62 The Uniqueness of the <iplicative

2-3.

Inverse

Once again we turn our attention to the problem o f matrix d i v i s i o n .

As we

have seen, rhts problem arises when we seek to solve a matrfx equation of the form

L e t us look a t a parallel equation concerning real numbers,

Each nonzero number a has a reciprocal l l a ,

B' .

which is often designated

Since multiplication of real numbers i s -1 commutative, it follows that a a = 1. Hence if a l a a nonzero number, then there is a number b, called the multiplicative inverse of a, such that e a l = 1.

Its defining property i s

-1

ab, = 1 = ba

Given an equation

us

to

ax =

C,

where

f i n d a solution for x;

a f 0,

(b = a

the multiplicative inverse

b

1.

enables

thus,

b(ax)

= bc,

( b a h = bc,

lx = bc, X

= h.

Now our question concerning division by matrices can be put in another way. A B

M,

i s there a

issatisfied? verse,

80

B

E

M for which the equation

-L

W e s h a l l e m p l o y themoreauggeativenotation A

that our question can be restated:

which the equation

If

for t h e i n -

Is there an element A - ~ E M

for

63 i s satisfied? s t a t e it

Since we shall often be using t h i s d e f i n i n g properw, let us

formally as a definition.

Definition 2-3. A

If

A E

M,

then an element

A-'

of

M

is an inverse of

provided

If there were an element

B corresponding

to

each element

A

f

M such

tht

AB = 1 s BA, then we could solve a l l equations of the form

AX = C, since we m u l d have

B(AX) = BC, (BA)X =

BC,

TX

BC,

-

X = BC, and clearly t h i s value s a t i s f i e s the original equation.

From the fact that there is a multiplicative inverse for every real number except zero, we might: wrongly infer a parallel conclusion for matrices. stated in Chapter 1, not a1 1 matrices

As

have inverses. h r knowledge that 0 has no inverse suggests that the zero matrix 0 has no inverse. This is true, since we have

-ox for all

X c

M, so that there

=

0

cannot be any

-ox = I. [set. 2-31

X

E

PI such that

92 Galois (1811-1832),

Group Theory.

to whom is c r e d i t e d the origin of the systematic study of

Unfortunately, Galois was killed in a duel at the age of 2 1 ,

i m e d i a t e l y a f t e r recording some of his most notable theorems.

Exercises 2-6 1.

Determine whether the following s e t s are groups under mulkiplication:

(b)

'I,

-I,

K,

-K,

where

2.

Show that the s e t a f a l l elements of

M

o f the form

c o n s t i t u t e s a group under multiplication.

3.

Show that the set of all elements of

[: :] constitutes a

,

where

t E

R.

s

e

M

R,

of the form

and

group under multiplicarion.

show t h a t the s e t

(A, A ~ A , ~ )

[see. 2-61

t

2

- s2 = 1,

93 Plot the corresponding points i n the

i s a group under multiplication.

plane.

5.

Let

Show that the a e t

1s this true i f

is a group under multiplication.

T

is

invertible

matrix? 6.

Show that the set: o f a l l elements of the form

[: :],

with

a

7.

a

and

b

R,

b E R,

and

ab

-

1,

If you p l o t all of the p o i n t s

is a group under multiplication. with

E

(a,b),

as above, what sort of a curve do you get?

Let

and let

H be the s e t of ell matrices of the form with

XI f yK,

x E R

and

y G

R.

Prove the following: (a)

The product of two elements of

(b)

The element XI +yK x

(c)

2

i s also an element o f

H.

is invertible i f and only i f - y

The s e t of a l l elements

under multiplication.

H

2

+o.

XI

+ yK

with

x

2

- y2

= 1 is a group

94 8.

If a set

G

of

2 x 2 matrices i s a group under multiplication, show that

each of the following s e t s are groups under multiplication:

9.

where 'A

(a)

{ A ~ ;A E GI,

(b)

(8-I AB: A E G ) ,

If a a e r

G

(a)

G

(b)

G

of

=

-

2 x 2 A

(BA: A

E

= transpose of

where B

A;

is a fixed invertible element of M.

matrices is a group under multiplication, show that E

GI,

GI,

B

where

is any fixed element o f

G.

Using the d e f i n i t i o n of an abstract group, demanstrare whether or not each

10.

of the following sets under the indicated operation is a group: (a)

the s e t of odd integers under addition;

(b)

the s e t

(c)

the s e e of the four fourth root8 of

R+

of posririve real numbers under multiplication;

1,

1

i, 4 , under

mu1 t i p l i e a t i o n ; (d)

the s e t of all integers of the form

3m,

where m

is an integer,

under addition.

11.

By proper application o f the four defining postulates of an abstract group, prove that i f a, b, and c are elements in a group and a o b = a o c, then

2-7

.

b =

c.

An Isomorphism between Complex Numbers and Efatrfces

It

i s true t h a t very many different kinds of algebraic systems can

expressed in t e r n of special collecrions of matrices. nature have been proved in modern higher algebra.

be

Wny theorems of thir

Without attempting any such

proof, we shall aim in the preaent s e c t i o n to demonstrate how the system of

complex numbers can be expressed in term of matrices. In the preceding section, several subaeta of the s e t of all

were d i s p l a y e d .

I n particular,

[Y" was considered.

I:

the s e t

*

2 x 2

matrices

Z of all matrices of the form

x E R and y E R ,

We shall exhibit a one--t-ne

correspondence between the a e t

95 of

all c 0 m p l e ~numbers, which we denote by

C,

and the s e t

2.

Thf s

one-to-one

correspondence would not be particularly significant if it d i d not preserve algebraic properties

- that

is,

if the sum of tw complex numbers b i d not

correspond to the sum of the corresponding two matrices and the product of tvo complex numbers d i d not correspond to the product: of the corresponding two

matrices.

There are other algebraic properties that are preserved in this

sense.

Usually a complex number is expressed in the form

where

1=

6 1 ,

x

E

R,

and y

h

R.

Thus, if

c

is an element of

C,

the

s e t of a l l complex numbers, we may write

L

The numeral

is introduced in order t o make the correspondence more apparent.

In order to exhfbit: an element: of

Z

i n similar

form, we must introduce the

special matrix

Note that

thus

The matrix

J

corresponds t o the number

equation

This enables us t o v e r i f y that

i,

*

which satisfies the analogous

which indicates that any element of

Z

may be written in the form

For example, w have

and

Now we can establish a correspondence between

the s e t of complex numbers,

2 , the s e t of matrices:

and

Since each element of of

C,

Z

C

is matched with one element of

is matched with one element o f

Z,

and each element:

C, we c a l l the correepondence o n e t r r o n e .

Several special correspondences are notable:

As s t a t e d earlier, it is interesting that there is a correspondence

between the complex numbers and

2 x 2

matrices, but the correspondence is not

particularly signiftcant unless t h e o n e t m n e matching is preserved in the operations, e s p e c i a l l y in addition and multiplication.

We shall now f o l l o w the

correspondence in these operations and demonstrate t h a t the one-tcrane property

is preserved under the operations. When two complex numbers are added, the real components are added, and the imaginary components are added.

Also, remember that the multiplication of a

matrix by a number is distributive; thus, far have

Hence we are able to show

a E il,

b

€

R,

and

A

E

M,

we

our o n p t o a n e correspondence under addition:

For example, we have (2

- 3i)

+

(21 - 35)

( 4 f li)

= 6 - 2 1

+

(41 + 13)

=

+ (27: + OJ)

-

61-2.T.

and

(3

- 2 i ) + (2 + O i )

=5-21

(33 f-3

-

25)

51-25,

Before demonstrating that the correspondence 513 preserved under multiplication, let us review for a moment.

An example w i l l suffice:

Generally, for multiplication, we have

If

IM

represent a complex number

as a matrix,

w e do have a s i g n i f i c a n t correspondence!

Mot only is there a o n e t m n e c o r

respondence between the elements of the two s e t s , but a l s o the correspondence

is onet-ne

under the operations of addition and multiplication.

The addftive and multiplicative identity elements are, respectively,

and

and for the ~ d d tive f inverse of

we have

Let uil now examine how the multiplicative inverses, or reciprocals, can be We have seen that any member of the s e t o f

matched.

2 x

2

matrices has a

multiplicative inverse i f and only i f for it the determinant function does not equal zero. x

2

+y

2

#

A E Z

Accordingly, if

0 , since

6(A)

= x2

+ y2

then there exlacs for A = XI

-

+ yJ.

A

i f and only if

M u we know that any

complex number has a multiplicative inverse, or reciprocal, if and only i f the complex nrrmber is not zero.

multiplicative inverse of and

y

are not bath 0 .

since x

E

R

and

y E R.

That is, i f

c

c

if and only i f

x

+ yi,

then there exists a

x

+ yi #

0,

which menne that

Thls l a equivalent t o saying that x2

+y

2

# 0,

For m l t i p l i c a t i v e inverses, i f

our correspondence yields

It i a now clear that the correepoadeace between C, nrrmbers, and

2,

a subset of all 2 x 2

matrices,

the s e t of complex

x

100 is preserved under the algebraic operations. saying that

C

and

2

AIL o f this may be summed up by

have i d e n t i c a l algebraic structures.

Another way of

C and Z are isomorphic. This word is derived from two Greek words and means "of the same form." Two number systems are

expressing t h i s i s to say that

isomorphic i f , f i r s t , there is a mapping of one onto the other that i s a o n e t o -

one correspondence and, secondly, the mapping preserves sums and products.

If

t w o number systems a r e isomorphic, their structures are the same; it is only

t h e i r terminology t h a t is d i f f e r e n t .

The world is heavy with examples

morphisms, some o f them t r i v i a l and some quite the o p p o s i t e .

of

is*

One of the simplest

is the isomorphism between the counting numbers and the p o s i t i v e integers, a subset of the integers; another fs that between the real numbers and the subset o f a l l compLex numbers.

a 4- Oi

(We should quickly guess that there is an

isomorphism between real numbers a1

a

and the set of all matrices of the form

+ OJ!) An example of an isomorphism that i s more difficult to understand is that

between real numbers and residue classes of polynomials.

We won't

t r y to explain

t h i s one; but there is one more fundamental concept that can be introduced here, as follows.

We have stated that the real numbers are isomorphic to a subset of the c plex numbers.

m

We speak of rhe algebra of the real numbers as being embedded in

the algebra of complex numbers.

In this sense, we can say that the algebra of 2 x 2

complex numbers i s embedded i n the algebra of

A l s o , we can

matrices.

speak of the complex numbers as being "richerf1than the real numbers, or o f the 2 x 2

matrices a s being richer than the complex numbers.

The existence of

complex numbers gives us solutions to equations such as

which have no solution i n the domain of real numbers.

that

Z

i s a proper subset of

M,

that: is,

example to illustrate the statement that

M

Z

C

It is o f course clear M and Z # M. Here i s a simple

is "richer" than

has f o r solution any matrix X =

[l;t

:],

t t R

[see. 2-73

and

f

0,

2:

The equation

101 as may be seen quickly by easy computation; and there are still other s o l u t i o n s .

On the other hand, the equation

has exactly two s o l u t i o n s among the complex numbers, namely c =-

c = 1

and

1.

Exercises 2-7 1,

Using the fol/loving values, show the correspondence under a d d i t i o n and

multiplication between complex numbere of the form of the form x 1

2.

and matrices

+ yJ:

- 1, x2 = 0,

(a)

x1 = 1, y1 =

(b)

X1

(c)

x1 = 0 , y l = - 5 ,

a

x f yi

= - 2;

and

y

1,

and

y2 =

x2=3,

and

y2=4.

31 Y1 = - 4 , xZ

ti

2

1;

Carry through, in parallel coluums as in the t e x t , the necessary computations t o establish an isomorphism between

R and the s e t

by means of the correapondence

3.

In the preceding exercise, an iaomorphi~mbetween matrices

was considered.

Define a function

f by

R

and the sets of

102 Determine which of the following statements are correct: f l x + Y ) = f(x)

(a)

4.

Is the

with

2-8.

set

G

and

a

+ f(y1,

of matrices

b

rational and

a

2

+ b2

-

1,

a group under multiplication?

Algebras

The

concepts of group, ring, and f i e l d are of frequent occurrence i n modern

algebra.

The study o f these systems is a study o f the arructures or patterns

that are the framevork on which algebraic operations are dependent. chapter,

r ~ ehave

In this

attempted t o demonstrate how these aame concepts describe the

structure of the

s e t of

2

X 2

matrices, which is a subset o f the s e t of all

rectangular matrices. N o t only have we introduced these embracing concepts, but we have exhibited

the ''algebra" of the s e t s . i n a Loose sense.

"Algebra" is a generic word t h a t i s frequently used

By technical definition, an algebra is a system that has tuo

binary operations, called "addition" and "mu1 tiplicarion," and also has

"multiplication by a number," that make it both a ring and a vector space. Vector spaces will be discussed in Chapter 4,and we shall see then that

the s e t of

2 x 2

matrices constitutes a vector space under matrix addition

and multiplication by a number. As you

Thus the

2

x 2 matrices form

yourself might conclude at t h i s t i m e ,

many possible algebras.

,this

an algebra.

algebra i a only one of

Some of these algebras are duplicate8 o f one another

in the aense that the b a s i c structure of one is the aame a a the basic structure

of another.

Superficially, they seem different because of the terminology.

When they have the same structure, t w o algebras are called isomorphic,

Chapter 3

MATRICES AND LINEAR SYSTEMS

3-1.

Eauivalent Systems

In this chapter, we shall demonstrate the use of matrices in the solution

of systems of linear equations.

We shall f i r s t analyze some of our present

algebraic techniques for s o l v i n g these systems, and then show how the same

techniques can be duplicated in terms of matrix operations. Let u s begin by looking at a system of three linear equations:

In our f i r s t s t e p toward finding the s o l u t i o n s e t of t h i s system, we proceed as follows:

M u l t i p l y Equation (1) by

Equation (1) by

-1

1

t o obtain Equation ( 1 ' ) ; multiply

and add the new equation t o Equation ( 2 ) t o obtain Equation

( 2 ' ) ; multiply Equation (1) by -2 and add the new equation to Equation (3) t o

obtain Equation ( 3 ' ) .

This gives the following system:

Before continuing, we n o t e t h a t what we have done is reversible. can obtain System I from System I1 as follows: to obtain Equation (1); add Equation

( 2 ) ; m u l t i p l y Equation (1')

(L' )

(2')

b h l t i p l y muation (1')

by 1

t o Equation ( 2 ' ) t o obtain Equation

by 2 and add t o Equation ( 3 ' ) t o obtain Equation ( 3 ) .

Our second etep is similar t o the first:

(1"); multiply Equation ( 2 ' )

In fact, we

Retain Equation ( 1 ' ) as Equation

by -1 to obtain Equation (2"); multiply Equation

by 3 and add the new equation to Equation (3') t o o b t a i n Equation ( 3 " ) .

This gives

Our t h i r d s t e p reverses the direction;

Mu1 t i p l y Equation (3") by - 1/8

to obtain Equation (3"' ) ; mu1 t i p l y Equation (3") by

3/8

(2") to obtain Equation (2"' ) ; multf p l y Equation (3") by Equation ( 1 " ) to obtain Equation (1"' ) We thus get

.

x - y

and add t o Equation

1/8 and add t o

+ o =-I,

O + y f 0 = 2 ,

O + O + z = - 1 .

Now, by retaining the second and t h i r d equatione, and adding the second equation to

the f i r s t , we obtain

or, in a more familiar

form,

In the foregoing procedure, we o b t a i n system I1 from system I, 111 from

11, fV from 111, and V from IV.

Thus

we

know that any set of values that

s a t i s f i e s system 1 must also satisfy each succeeding system; in particular, from eystem V we f i n d that any

(K, y , z)

that s a t i s f i e s

I must be

Accordingly, there can be no other s o l u t i o n of the original system I; i f there

is a s o l u t i o n , then t h i s is it.

,

But do the values sys terns

of linear equations

2,

-

we

have already p o i n t e d

actually s a t i s f y system out

I?

For our

that sys tern I can be

obtained from system 11; similarly, I1 can be o b t a i n e d from 111, 111 from IV, and I V from V.

Thus the s o l u t i o n of

V, namely

(1, 2 , - 1

Of course, you could verify by direct substitution t h a t

s a t i s f i e s I. ( 1 , 2 , -1)

s a t i s f i e s the s y s t e m I , and actually you should do this to guard against corn-

putational error.

But it is useful to note explicitly t h a t the ateps are

reversible, so t h a t the systems I, 11, 111, IV, and V are equivalent: in

accordance with the following definition: D e f i n i t i o n 3-1.

Two systems of l i n e a r equations are said to be equivalent

if and only if each s o l u t i o n

of

either system is also a solution of the other.

We know t h a t the foregoing systems I through " are equivalc

s t e p s we have taken are reversible.

..: because the

In f a c t , the only operations we have per-

formed have been o f the f o l l o w i n g sorts: A.

Multiply an equation by a nonzero number.

B.

Add one equation to another.

Reversing the process, we undo the addition by subtraction and the multiplication by d i v i s i o n .

Actually, there is another operation we shall sometimes perform in our systematic solution of systems of linear equations, and i t a l s o is r e v e r s i b l e :

C.

Interchange two equations.

Thus, in s o l v i n g the system

our first step would be to interchange the f i r s t two equations in order to have a leading

coefficient d i f f e r i n g from zero.

In the present chapter we shall investigate an orderly method of elimina-

t i o n , without regard t o the particular values o f the coefficients e x c e p t that we shall a v o i d d i v i s i o n by

0.

Our method will be especially useful in dealing

with several s y s t e m s in which corresponding coefficients of the variables are

106 membere are d i f fereat

equal while the righ-ad

-

a situation that of ten

occurs in industrial and applied ecientific problems.

You might use the procedure, for example, i n "programming, a method, or program, for solving a system of

'I

i .e

., deviaing

Linear equations by mane o f a

modern electronic computing machine.

Exercises S 1 1.

Solve the following ayaterrm of equation@: (a)

(e)

+ 4y 5x + 7 y

3x

--

x+Zy+

4, 1;

z-3w-2,

w - 7 ,

y - 2 2 -

z

- 2w u

2.

(f)

-

OX

* 0,

+ ly + oz

+OW

b,

Ox+Oy+la+Ov=c,

3;

Ox f Oy + 0 z

+ lw = d .

Solve by drawing graphs:

(b)

3x

-

-

y =

5x + 7 y 3.

-

lxfOy+Oz+Ow=a,

11, 1.

Perform the following matrix multiplications:

4. Given the following systems A and B, obtain B f r o m A by s t e p s consisting either of w l t i p l y i n g an equation by a nonzero oonstant OF of adding an arbitrary multiple of an equation t o another equation:

107 Are the two systems equivalent?

Why o r why not7

The solution ser of one of the following sys terns of linear equations is

j,

empty, while the other s o l u t i o n s e t contains an i n f i n i t e number o f See i f you can determine which i s

solutions.

which, and g i v e three

particular numerical solutions for the system that does have s o l u t i o n s : (a)

x+2y-

z=3,

x -

z = 4 ,

4x

2

.

-

y +

y f

22

=t

14;

(b)

x -I-2y x -

4x

-

- z = 3,

y + y

z = 4 ,

+ 22

115.

Formulation in Terms of Matrices

In applying our method t o the s o l u t i o n o f the original system of Section

3-1, namely to

we carried o u t j u s t two types of algebraic operations in o b t a i n i n g an equivalent

sys tern: A.

Multiplication of an equation by a number o t h e r than

B.

A d d i t i o n of an equation to another equation.

0.

We noted that a third type o f operation i s sometimes required, namely:

C.

Interchange of two equations.

This third operation is needed if a c o e f f i c i e n t by which d i v i d e is

* otherwise

would

0, and there is a subsequent equation in which the same variable

has a nonzero c o e f f i c i e n t . The t h r e e foregoing operations can, in e f f e c t , be carried out through

matrix operations.

Before ve demonstrate this, we shall see how the matrix

notation and o p e r a t i o n s developed i n Chapter 1 can be used t o write a system of linear equations i n matrix form and t o represent the steps in the s o l u t i o n . Let us again consider the system w warked with in Section 3 1 :

We may d i s p l a y the detached coefficients o f

x, y,

and

as a matrix

z

A,

namely

Next, l e t

US

consider the matrices

.=[!I

the entries of

X

are the v a r i a b l e s

and

I'[

B = x , y, z ,

members of the equations we are considering.

and of

;

B are the right-hand

By the definition o f mgtrix

multiplication, we have

which is a

3 x 1

matrix having as entries the l e f e h a n d members of the

equations of our Linear s y s rem. NOW the equation

thar is,

is equivalent, by the definition of equality o f matrices, to the entire sys tern o f linear equations.

It

i s an achievement not t o be taken modestly

thar re are

a b l e to consider and work with a large sys tern of equations in tern o f such a

log s i m p l e representation as is exhibited in Equation (1).

Can you see the pattern

t h a t is emerging? In passing, let us note that there is an i n t e r e s t i n g way of viewing the matrix equation

where

is a given

A

3 x 3 matrix and X

Y are variable

and

3 x 1 colurrm

We recall that equations such as

matrices.

and

y d e f i n e f u n c t i o n s , where numbers

the range.

x

=

sin x

of the domain are mapped into numbers

We may also consider Equation ( 2 ) as defining

case the domain c o n s i a t s of column matrices

matrices

Y;

X

a

and the range consists of column

For example,

the matrix equation

and a range of

3 x 1

of

function, but in this

thus we have a matrix function o f a matrix variable!

defines a function with a domain of

y

matrices

3 x 1 matrices

Thus with any column matrix of the form (3),

the equation associates a

UO column matrix of the form ( 4 ) . Looking again at the equation

where

is a g i v e n

A

a variable

matrices

X

3

3 x 3 matrix,

B

is a given

x 1 matrtx, we note that here we

3 x 1 matrix, and

X

have an inverse question:

( i f any) are mapped onto the particular

B?

For the case we

is What

have

been considering,

we

found in Section 3 1 that the unique s o l u t i o n is

We

shall consider some geometric aspects

of

t h e matrix-function p o i n t of

view in Chapters 4 and 5. We a r e

now ready to look again at

the procedure of Section 3-1, and restate

each system in terms of matrices:

[i 1 -1

The t-eaded

arrows

[I [j]" =

indicate Lhat, as w saw in Secrion 3-1,

tho matrix

equations are equivalent,

In order t o see a little more clearly what has been done, let us look o n l y [sec. 3-21

a t the

f ixst and last equations.

The first is

the last is

[g g g] [i] - [.i]. Our process has converted the c o e t f i c i e n r matrix

A

into the i d e n t i t y matrix

I. Recall that

Thus t o bring about this change we m u s t have completed operations equivalent to multiplying on the l e f t by

A'.

In brief, our procedure a m u n t s

on the left by

t o multiplying each member of

A-l,

to obtain

Let ua note how far we have come.

By introducing arrays of numbers as

o b j e c t s in their own rfght, and by defining s u i t a b l e algebraic operations, we

are able to write complicated systems of equations in the simple fonn

This condensed notation, so similar to the long-familiar

which

we

Learned t o solve ''by d i v i s i o n , " indicates for us our s o l u t i o n process;

namely, multiply both sides on the left by the reciprocal or inverse o f

A,

obtaining the s o l u t i o n

This similarl ty be tween

and

should nor be pushed

too

f a r , however.

There is only one real number,

0,

that

has no reciprocal; as we already know, there are many matrices that have no multiplicative inverses.

Nevertheless, we have succeeded in our aim, which

perhaps the general aim of mathematics :

to

i s

make the complicated simple,by

discovering i t s pattern.

Exercises 3 2 1.

Write in matrix £ o m : (a)

4x

- 2y

f 72 = 2 ,

3x4- y + 5 z = - I , 6y-

2.

(b) x

$

y = 2,

x - y = 2 .

z=3;

Determine the systems o f algebraic equations t o which the following matrix equations are equivalent:

3.

Solve the following s y s t e m of equations; then restate your w r k In matrix form:

4.

(a)

Onto what vector

map the vector

5.

Let

A =

domain

of

[;]

does the function defined by

Y

=

[i] ,

?

(b)

-

What vector

[Y~] the function d e f i n e d by

[al a2 a) a&]

Y

,

I:[

does i t map onto the

ai, xi, y1 E R.

Discuss the

Define the inverse function, if any.

3

Inverse of a Matrix

In Section 2-2

we

wrote a system of linear equations in matrix form,

and saw that solving the system amounts t o determining t h e inverse matrix

-1

A

,

if it e x i s t s , since then ve have

whence

Our work involved a series of algebraic operations; l e t us learn how to duplicate

t h i s work by a series of matrix alterations.

TO do t h i s ,

we suppress the column

matricea in the scheme shown in Section 3 2 and look a t the coefficient matrices

u4 on the Left:

Observe what happens if ue s u b s t i t u t e "row" for "equation" in the procedure o f

Section 3-1 end repeat our steps.

In the first s t e p , we mu1 tiply row 1 by -1

and add the result t o row 2 ; multiply row 1 by -2 and add t o row 3 . m u l t i p l y row 2 by 3 and add t o row 3; multiply row 2 by -1.

row 3 by

3/8

and add t o row 2 ; multiply row 3 by

multiply row 3 by

- L/8. b a t ;

118

we add row 2 t o row 1,

Second, we

Third, ue multiply

and add to row 1;

Through "row operations"

we have duplicated the changes i n t h e c o e f f i c i e n t matrix as w proceed through

the series of equivalent

system^.

The three algebraic operations described in Sec tlon 3-2 are peralleled by three matrix row operations: Definition 3-2.

The three row operations,

Interchange of any tw r o w , Multiplication of a l l elements of any row by a nonzero constant,

Addition o f an arbitrary multiple of any row to any othet row, are c a l l e d elementary row operations

on

a matrix.

the exact relationship between row operations and the

In S e c t i o n 3-5,

operations developed i n Chapter 1 w i l l be demonstrated.

Earlier we d e f i n e d

equivalent systems o f linear equations; i n a corresponding way, we now define equivalent matricee

.

Definition 3 3 .

Two matrices are said to be row equivalent i f and only if

each can be transformed into the other by means of elementary row operations. We

now turn our attention t o the right-hand member

A t the moment, the right-hand

of

the equation

member consists solely of t h e matrix

B,

we wish temporarily t o suppress just as w temporarily suppressed the

X

in considering the left-hand member.

matrix for the right-hand member.

which

matrix

Accordingly, ue need a coefficient

To obtain this, we use the i d e n t i t y matrix

[seem

3-31

to write the equivalent equation AX =

IB.

Now our process converts this to

which can be expressed e q u i v a l e n t l y as

When we compare EquatFon (1) and Equation ( 2 ) we notice t h a t as

I

goes to

A

A-l. Might it be t h a t the row operation*, which convert

when applied t o the left-hand member, w i l l convert t o the right-hand member?

Let us t r y .

goes t o

A

I,

into

I into A-Iwhen applied

For convenience, w e do t h i s in paralLel

columns, giving an indication in parentheses of how each new row is obtained:

I

4 --

L

O

O

0

0

1

left-hand Fnverse f o r

A,

8

4

4

8

8

To demonstrate that

i s a

it is necesaary to show that

are asked to verify this as an exercise, and also to check that AB = I, -1 of A . In Section 3-5 we shall thus demonstrating that 3 is the inverse A

You

see why i t is that

AB

-

1

follows from

BA = I

for matrices

B

that are

obtained in this way as products of elementary matrices.

We now have the f o l l o w i n g rule for computing the inverse A,

i f there is one:

A-'

of a matrix

Find a series of elementary row operations rhat convert

I; the same series of elementary row operations will convert the i d e n t i t y matrix I into the inverse A-'.

A i n t o the i d e n t i t y matrix

When we s t a r t the process we may n o t know if the inverse e x i s t s .

need n o t concern us at the moment.

that i s , i f

If the application of the r u l e i s successful,

then we know that -1 s e q u e n t s e c t i o n s , w e s h a l l d i s c u s s w h a t h a p p e n s when A A

is converted into

This

I,

A'

exists.

In sub-

doesnotexistandwe

a h a l l a l s o demonstrate that the row operations can be brought about by means of the matrix operations rhat were developed i n Chapter 1.

Exercises 3 3 1.

Determine the i n v e r s e of each of the following matrices through row operations.

(Check your answers.)

2.

Determine the inverse, if any, of each of the f o l l o w i n g matrices:

3.

S o l v e each of the following matrix equations:

(a'

4.

[:: 5

:I [;I - [a]

1 -1

,

[: :

Solve 1 -1

x

u

m

r

3

1 - 6 -

3

5.

$1

I -1

5

Perform the multiplications

-9

-

=

[i] ,

6.

M u l t i p l y both member6 of the matrix equation

on the left by

and use the reaulr to solve the equation. 7.

Solve:

8.

Solve:

.

U9 Linear Systems of Equations I n Sections 3 2 and 3 3 , a procedure for ~ o l v i n ga mystem of linear equations,

was presented.

exists

.

The method produces the multiplicative inverse A-I

i f it

In the p r e ~ e n tsection we shall consider systems of linear equatione in general.

Let us begin by looking at a simple illustration. Example 1.

Consider the system of equations,

We start in parallel eolunu~s, thus:

Proceeding, we arrive after three steps a t the following:

If we multiply these two matrices on the right by the matrices

we obtain the s y s t e m

There is no mathematical loas i n dropping the equation

0 = 0

from the system,

which then can be written equivalently as

Whatever value Is given t o and

y

z,

t h i s value and the corresponding values of

determined by these equations s a t i s f y the original system.

x

For example,

a feu solutions are sham in the following table:

Example 2 .

Now consider the systems

If we s t a r t in p a r a l l e l columns and proceed as before, we obtain verify)

( a s you should

Multiplying these matrices on the right by the column matrices

we o b t a i n the systems

If there were

a solution o f t h e original system, it would

this l a s t system, which is equivalent to the f i r s t .

no

tains the contradiction 0 = - 1; hence there is

Do you have

an

also be a solution of

But the l a t t e r system con-

s o l u t i o n of either system.

i n t u i t i v e geometric notion of what might be going on in

each of the above systems?

Relative to a 3dimensional rectangular coordinate

system, each of the equations represents a plane.

actually intersect in a line.

Each pair of these planes

We might hope that the three lines of intersection

(in each system) would intersect in a point.

In the f i r s t system, however, the

three lines are coincident; there ts an entire "line" of solutions.

On

the

orher hand, in the second system, the three lines are parallel; there i~ no point that lies on all three planes. 3-2

In the example worked out in Sections

and 5 3 , the three lines intersect in a s i n g l e point. How many possible configurations, as regarda intersections, can you l i s t

f o r 3 ~ l a n e s ,n o t n e c e s s a r i l y d i s t i n c t from one another?

They might, for example,

have exactly one point in common; o r two might be c o i n c i d e n t and the t h i r d d i s r i n c t from but parallel t o them; and so on.

There are systems of linear

equations r b t correspond t o each o f these geometric situations.

Hers are two additional systems that even more obviously than the sys tern in Example 2, above, have no solutions:

n u s you see

that the number o f variables as compared with the number oL equations

does n o t determine whether o x n o t there i s a solution. [sec. 3-41

I22 The method we have developed for solving linear systems I s routine and can be applied

ables

.

to

sys terns o f any number of linear equations i n any number of vari-

Let us examine the general case and see w h a t can happen.

have a Bystem of m

linear equations in

'If any variable occurs with coefficient

n

0

Suppoae we

variables:

in every equation, we drop it.

If

a coefficient needed as a d i v i s o r is zero, we interchange our equatlons; ve might

even interchange some of the terms in a l l the equations for the convenience of

getting a nonzero coefficient in leading position.

Otherviae we can proceed in

the customary manner un t i 1 there remain n_o equations in which any of the variable6 have nonzero c o e f f i c i e n t s , Eventually we have a syatem like this:

X1

linear

terms

in variables other t b n xl

+

... 5

5

PI,

a

B2,

tl

X2

+

and (possibly) other equations i n which no variable appears.

Either all of these other equations { i f there are any) axe of the form

in which case they can be disregarded, or there is a t least one of them of the form

i n which caee there i r a contradiction.

If there is

a contradiction, the contradiction was present in the original

system, which means there is no solution s e t .

I23 In case there are no equations of the form

..,%

we have two po~sibilities. Either there are no variables other than xl,. which means chat the system reducee t o

..,%

and has a unique solution, or there really are variables other than xl,, to

which we can assign arbitrary values and obtain a family of s o l u t i o n s as i n

Example 1 , above

. Exercises 3-4

1.

(a)

List all possible configurations, a8 regards intersections, for 3 d i s t i n c t planes

(b)

.

L i s t a l s o the addf tional possible configurations i f the plane8 are

al loved to be coincLdent 2.

.

Find the ~ o l u t i o n s ,i f any, of each of the following systeme of equations: (a)

x

+

y

+ 22

= 1,

Z

= 3,

2x f

3x

(b)

+ 2y + 42

4

4;

x +

y +

x +

y+22=7, y +

z-6,

2-1;

(d)

2v+ v -

4v

x +

y +

t P O ,

x + 2 y +

2-0,

- x

v-

X

+ 5y + 33 = f

y -

1,

2 - 2 ;

l.24 y +

z + w = 2 ,

x + 2 y f

z - w = - 1 ,

2x+

(e)

4 x + 5 ~ + 3 2 - W = 0 .

3-55.

Elementary Bow Operations three types o f algebraic operations were l i s t e d as being

In Section >I.,

involved i n the s o l u t i o n o f a linear system o f equations; in Section 3 4 , three types of row operations were used when we d u p l i c a t e d the procedure employing

In t h i s section

matrices.

this earlier w r k ,

- in

we shall show how matrix multiplication underlies

f a c t , how t h i s basic o p e r a t i o n can duplicate the row

operations.

Let us start by looking a t whit happens if

row operations on the i d e n t i t y matrix by a nanzero number

X

Let

represent any matrix o f this type.

J

perform any one o f the three

of order 3 ,

I

row o-f

n,

Me

we have

a

First if we mu1 t i p l y a

matrix of the form

Second, i f w e add one row of

I

another, we obtain

or one o f t h r e e other similar m a t r i c e s . any matrix of' this t y p e .

(denoted by

L)

(What are they?)

Let

K

arand for

Third, i f we interchange tuo rows, we form matrices

o f the form

Matrices of these three types are c a l l e d elementary matrices. D e f i n i t i o n 3-4. An elementary matrix is any square matrix obtained by

performing a single elementary raw operation on the identity matrix.

to

Each elementary matrix E ( t h a t is, each -1 t h a t is, a matrix E such t h a t

3, K,

or

L)

has an inverse,

For example, the inverses of the elementary matrices

are the elementary matrices

respectively, as you can v e r i f y by performing the multip~icatfonsFnvolved. An elementary matrix is related to its i n v e r s e i n a v e r y simple w a y .

J

example,

J-1

was obtained

by multiplying a row of the i d e n t i t y matrix by

is formed by dividing the same row of the i d e n t i t y matrix by

aense

-1

J

"undoes" whatever was done to o b t a i n

conversely,

J

will undo J-l.

J

n.

For

n;

In a

from the i d e n t i t y matrix;

Hence

The product of two elementary matrices also has an inverse, as the following theorem i n d i c a res Theorem 3-L, and

B-',

Proof.

and

.

I f square matrices

then AB

We have

A

has an inverse

and

B

CAB)-',

of order

namely

n

have inverses

126 so that

-1 -1 B A

is the inverse o f

You will recall that, for

AB

2 x 2

by the d e f i n i t i o n of inverse. matrices, t h i s same proof was used in

establishing Theorem 2.8.

-1 A

...,

Corollary 3-1-1. If square matrices A , B , K of order -1 -1 , B ,, , , then the product AB. * K has an inverse

-

.

(

n M

a

have inverses

*K)-I,

namely

The proof by mathematical induction is left: as an exercise. Corollary 3-1-2.

If

E1,E2,.

..,%

are elementary matrices of order

n,

then the matrix

has an inverse, namely

This follows inmediately from C o r o l l a r y 3-1-1

and the fact that each

elementary matrix has an i n v e r s e .

The primary importance of an elementary matrix r e s r s on the following property.

If an m X n

elementary matrix

E,

matrix

A

is multiplied on the left by an

m x n

then the p r o d u c t is the matrix obtained from A

the row operation by which the elementary m a t r i x was formed from

I.

by For

example,

You should v e r i f y these and o t h e r similar left multiplications by elementary

matrices t o familiarize yourseL f with the patterns.

Any elementary row operation can be performed on an

Theorem 3-2. A

matrix of order

by multiplying

A

m x n

on the left by the corresponding elementary matrix

m.

The proof is left as an exercise.

Multiplications by elementary matrices can be combined. add

-1

times the first r o w to the second row, we would multiply

left by the product of elementary matrices of the type

Note that in ;I'

J multiplies the first row by

-1;

A

on t h e

J-'KJ:

i t i s necessary t o multiply by

order to change the first row back agafn to its original form after

adding the f i r s t row to the second. to

For example, to

the third, we would multiply

A

Similarly, to add

-2

times the f i r s t row

on the left by

To perform both o f the above steps at the same time, we would multiply

A

on

the left by

Since

M1

is the product of two matrices that are themselves products of

9 is the product o f elementary matrices. By Corollary of 9 is the product, in reverse order, o f the corresponding

elementary matrices,

the inverse

3-1-1,

inverses of the elementary matrices. Now our f i r s t s t e p in the s o l u t i o n of the system of linear equations at

the start of this chapter corresponds p r e c i s e l y t o m u l t i p l y i n g on the left by the above matrix left by

5'

M1.

If we multiply both sides of

System

I on page 103 on the

we obtain System 11.

For the second, t h i r d , and fourth s t e p s , the corresponding

matrix multipliers m u s t be r e s p e c t i v e l y

Thus multiplying on the left by

M2

has the effect of leaving the first row

unaltered, m u l t i p l y i n g the second row by

-1,

and adding 3 times the second

row to the t h i r d . L e t us now take advantage o f the associative law for the nulriplication of matrices t o form the product

We recognize the inverse of t h e orLginal c o e f f i c i e n t matrix, as determined in S e c t i o n 3-3,

Theorem 3-3.

If

BA = I ,

where

i s a product o f elementary matrices, t h e n

B

A3 = I,

SO

that

3

-

Proof.

is the inverse of A .

By Corollary 3-1-2,

B has an inverse 3-1 BA = I {see. 351

.

Now from

we get

whence

so that

and

Exercises 3-5

A,

1

Find matrices

2.

Express each of the following matrices as a product of elementary matrices: (a)

1

B,

0

and

-J

[:1;

0

a

C

such t h a t

3.

Using your answers

to

Exercise 2 , form the inverse o f each of the given

matrices.

4. Find three

4 x 4

matrices, one each of the t y p e s

3,

K,

and

I,

that

w i l l accomplish elementary row transformations when a p p l i e d as a left

multiplier to a 4 x 2

matrix.

5.

Solve the following system of equations by means of elementary matrices:

6.

(a)

Find the inverse of the matrix

(b)

Express the inverse a product of elementary matrices.

(c)

Do you think the answer t o Exercise b(b) is unique?

Why or why not?

Compare, in class, your answer with the answers found by other members

of your class.

7.

Give a proof of Corollary 51-1 by mathematical induction.

8.

Perform each of the following multiplications:

m 9.

State a general conjecture you can

make on the basis of your solution of

Exercise 8.

- 5 - 3-

In this chapter, we have discussed rhe role of matrices in finding the solutlon s e t of a system of linear equations.

We began by Looking at the

fami liar algebraic methods of obtaining such solutions and then learned to duplica re this procedure using matrix notation and r o w operations

.

The intimate

connection between row operations and the fundamental operation o f matrix multiplication was d e m n s t r a t e d ; any elementary operation is the reeult of l e f t m u l t i p l i c a t i o n by an elementary matrix.

Ue can obtain the s o l u t i o n set, if it

e x i s t s , to a system of l i n e a r equations e i t h e r by algebraic methoda, o r by row o p e r a t i o n s , or by multiplication by elementary matrices.

Each of these three

procedures involves steps t h a t are reversible, a condition t h a t assures t h e equivalence of the sys terns. Our work with systems of linear equationa led as to a method for producing

the inverse o f a matrix when i t exists. that converts

a

matrix

The i d e n r i c a l sequence

o f rou operations

into the identity matrix will convert the identity

A

A, namely A-l. The inverse is particularly h e l p f u l when-we need to solve many systems of l i n e a r equations, each possessing the

matrix into the inverse of same c o e f f i c i e n t matrix

but d i f f e r e n t right-hand column matrices

A

B,

Since the matrix procedure 'diagonalized' the c o e f f i c i e n t matrix, the method is o f t e n called the "diagonalization method."

Although ve have not

previously mentioned it, there i s an a l t e r n a t i v e method for s o l v i n g a Linear system that

is o f t e n more u s e f u l when dealing w i t h a s i n g l e system.

In this

alternative procedure, ue apply elementary matrices to reduce the system to the

form

as in Sya tern

obtained.

III in Section 3-1,

This value o f

to determine the value o f

from which the value for

z

can readily be

can then be subs ti t u t e d in t h e next to Last equation

z

y,

and so on.

An examination of the c o e f f i c i e n t

132 matrix displayed above shows clearly why t h i s procedure is called the "trtangularization method On

."

the o t h e r hand, t h e diagonalizarion method can be speeded up to bypass

the triangularized matrix of c o e f f i c i e n t s in ( I ) , or in System 111 of S e c t i o n

3-1, altogether.

Thus, a f t e r p i v o t i n g on the c o e f f i c i e n t of

x

in the System

I,

to o b t a i n the s y s t e m

we could next p i v o t completely on

and then on the coefficient of

z

the coefficient of y

to obtain the system

to g e t the system

IV'

which you will recognize as the system V of Section 3-1,

It should now be plain that the routine diagonal i z a t i o n and triangularizat i m methods can be a p p l i e d to sya terns o f any number o f equations in any number of

variables and that the methods can be adapted to machine computations.

Chapter 4 REPRESENTATION OF COLUMN MATRICES AS GEOMETRIC VECTORS

4-1.

The Algebra o f Vectors In the present c h a p t e r , we shall develop a simple geometric representation

for a special class o f matrices - namely, the s e t o f column matrices with two e n t r i e s each.

The familiar algebraic operations on this s e t of matrices

will b e reviewed and also given geometric interpretation, which w i l l lead

to

a

deeper understanding o f the m e a n i n g and implications of the algebraic concepts,

By d e f i n i t i o n , a column vector of order

2 x 1

is a

2

matrix.

Consequent-

l y , usfng the rules o f Chapter 1, we can add two such vectors or m u l t i p l y any one of them by a number.

The s e t of column vectors o f order

2

has, i n f a c t , an

a l g e b r a i c structure with propertfes that were largely explored i n our study o f

the r u l e s o f o p e r a t i o n with matrices,

In the f o l l o w i n g pair o f theorems,

we sutmnarize what we already know con-

cerning the a l g e b r a o f these v e c t o r s , and i n the next section we shall b e g i n the interpretation o f that algebra i n geometric terms. Theorem 4-1.

number, and Let

Let: A

V

and

W

be column vectors of order

be a square matrix o f order

V

are each column vectors of order

+ W,

rV,

and

2.

2,

Then

AV

2.

For e x a m p l e , i f

V =

then

and

[:I

,

W =

]:[

, r

-

4,

and

A =

[i-:]

let

r

be a

19 Ler V,

Theorem 4-2. and

W, A

be numbers, and let

s

U

and

B

and

be column vectors o f order

2,

let

be square matrices of order

2.

Then

r

a l l the following laws are valid.

I. Laws for the addition

11.

(a)

V + W P W + V ,

(b)

(V f

(c)

v

+ 0 = v,

(d)

V

+ ( 4 )= 2.

o f vectors:

W) + U = V

+ (W

f

U),

Laws for the numerical multiplication o f vectors:

+ W)

= rV f rW,

(a)

r(V

(b)

~(BV)= (TS)~,

(c)

(r

+ s)V

(dl

ov

= 0,

(e)

lv =

(£1 rg

a

rV

+ sV,

-

=

V,

0.

III. Laws f o r the multipLication of vectors by matrieea: (a) A(V

+ W) + B)V

= AV = AV

+ AM, + BV,

(b)

(A

(c)

A(BV) = (AB)V,

(dl

02V =

(e)

IV = V,

(f)

A(rV) = (KA)V = r(AV).

2,

In Theorem 4-2, 0 denotes the column vector of order square matrix of order

2,

all of whose entries are

2,

and

O2

the

0.

Both of the preceding theorems have already been proved f o r matrices.

~'ince

column vectors are merely special types of matrices, the theorem as stared must

likewise be true.

They would a l s o be true, o f course, i f

3 or by a general

n,

of order

n

2 were replaced by

throughout, with the understanding that a column vector

is a matrix of order

n

x 1.

Exercises 4-1

1.

Let

let r = 2

and

a=-I.;

and l e t

Verify each of the laws s t a t e d in Theorem 4-2

for this choice o f values

f o r the variables. 2.

Determine the vector

V

such that

AV

- AW = AW

+ BW,

where

3.

Determine the vector

V

such t h a t

LV

+ 2W

+ BV,

if

4.

Find

5.

Let

V,

Evaluate

i F

= AV

Using your answers to parts (a) and ( b ) , determine the e n t r i e s of

(e)

if, f o r every vector

A

and

n

theorem for

A

State

is

square matrix of

a

stands f o r any column vector o f order

V

n = 3.

Try to prove the theorem for a l l

if, for every vector

V

of order

n.

Prove rhe new

n.

2,

your result as a theorem.

Restate the theorem obtained in Exercise 7 if A

n

and

n = 3. 9.

A

Using your answers to parts (a) and (b) of Exercise 5 , determine the entries of

8.

2,

Restate the theorem obtained in Exercise 5 if order

7.

of order

S t a t e your r e s u l t as a theorem.

(d)

6.

V

V

stands f o r any column vector of order

T r y to prove the theorem for all

is a square matrix of order

n.

Prove t h i s theorem f o r

n.

Theorems 4-1 and 6 2 surmnarize the p r o p e r t i e s of the algebra of column vectors with

2

entries.

S t a t e two analogous

properties of the algebra of row vectors with

theorems s u m r i z i n g the 2

entries.

'1

Show that the

t w o algebraic structures are isomorphic.

4-2.

Vectors and Their Geometric Representation The n o t i o n of a vector occurs frequently i n the physical sciences, where a

I

v e c t o r ia o f t e n referred to as a quantity having b o t h l e n g t h and d i r e c t i o n and

accordingly i s represented by an arrow.

Thus force, v e l o c i t y , and even dis-

placement are vector q u a n t i t i e s .

Confining our attention to the coordinate plane, l e t u s invesrigate t h i s I

i n t u i t i v e notion of vector and see how these physical or geometric vectors are related to the algebraic column and row vectors o f S e c t i o n 4-1.

An arrow in the plane is determined when the coordinates of its & and i J the coordinates of its head are g i v e n .

is shown in Figure 4-1.

Thus the arrow

A1

from ( 1 , Z )

to

(5,4)

Such an arrow, in a given p o s i t i o n , is called a Located

I

Figure 4-1.

.

.

. .

.

. * X

Arrows in the plane.

yeetor; its r a i l is called the initial p o i n t , and its head the terminal point (or end point), of the located vector.

h second l o c a t e d vector

and terminal p o i n t (2,5),

i n i t i a l point (-2,3) A located vector

A

A2,

with

i s also shown in Figure 4-1.

may be described briefly by

giving f i r s t the co-

ordinates of its i n i t i a l p o i n t and then the coordinates of i t s terminal p o i n t ,

Thus the located vectors in Figure 4-1 are A1:

(1,2)(5,4)

The h ~ r f z o n t a lrun, or

and its vertical rise, or

x

y

and

component, of

+: { - 2 , 3 ) ( 2 , 5 ) Al

is

component, is

Accordingly, by the ethagorean theorem, its length i s

a

'Its d i r e c t i o n is determined by the angle e x

that it makes w i t h the p o s i t i v e

direction:

sin

Since cos

sin

and

0

is the cosine of the angle that

9

makes with the

A1

are called the d i r e c t i o n cosines of A

8

y

direction,

1'

You might wnder why we d i d n o t wrlte simply tan 8 =

instead of the equations f o r value of

cos 8

and

2

1

T=T

s i n 8.

The reason is t h a t while the

determines slope, it does not determine the d i r e c t i o n of

tan 9

A1.

is the angle f o r t h e located vector (5,4)(1,2) opposite to A1,

Thus if

tan

fl and

but the angles

0

0

=

-2

=

12

they terminate in directions differing by Now the

x

and

tan,;

=

from the positive n

.

x

d i r e c t i o n cannot be equal s i n c e

components of the second located vector

y

then

A2:

(-2,3)

in Figure 4-1 are, respectively,

(2,S)

2

so that

AL

and

A2

- (-2)

have equal

= 4 x

and

5

- 3 = 2,

components and equal

y

sequently, they have t h e same length and the same direcrion.

components ; con-

They are not in

the same position, so of course they are not the same located vectors; b u t since in dealing w i t h vectors we are especially interested in length and direction we say that they are equivalent.

Definition 4-1.

Two located vectors are said t o be equivalent if and only

if they have the same length and the same direction.

P in the plane, there is a located vector (and to A2) and having P as i n i t i a l p o i n t . To determine

For any prescribed p o i n t

equivalent t o

A1

u9 the coordinates o f the terminal point, you have only t o add the components o f

A1

t o the corresponding caordinates o f

P: (3,-7),

P. Thus for the i n i t i a l

point

the terminal p o i n t is

so that the located

vector i s

You might plot the initial point and the terminal p o i n t of A

3

check that it actually is equivalent t o A1

In general,

and to A 2 ,

we denote the located vector

and terminal point

in order t o

A

with f n i t i a l point

(x

l'Y1)

by

(x2,yZ)

A: (x~'Y~)(x~*Y~) '

Its

x

components are, respectively,

and y

xZ-xL

and

Y2

- Y1-

Its length is

If r the

0, x

then its direceion l a determined by the angLe that i t makes w i t h

axis:

If r = 0,

we find f t convenient to say that the vector is directed In

any direction we please.

A s you w i l l s e e , this makes it possible t o s t a t e

~everaltheorems more simply, without having t o mention special cases. example, this

null

For

vector is both parallel and perpendicular t o every direction

i n the plane! A second located vector

is equivalent t o as

A,

A

if and only if it has the same length and the same direction

or, what amounts to the same thing, if and o n l y if it has the same c

ponents as

m

A:

For any given p o i n t

(xO,yO),

is equivalent t o

and haa

A

the located vector

(xO,yO) as its i n i t i a l point.

It thus appears that: the located vector

is determined except for its

A

p o s i t i o n by its components

a

= x2

- X1

and

b

a=

YZ - Y l -

These can be considered as the entries of a column vector

In t h i s way, any located vector for any given point

P,

A

determines a column vector

as the components of a located vector

locater vector

A

V

the entries of any column vector with

A

is said to represent

A colunm vector is a "free vector''

P

Conversely,

can be considered

as i n i t i a l p o i n t .

The

V. in the sense t h a t it determines the

components (and therefore the magnitude and direction), but: of

V.

any Located vector that represents it.

In particular,

the column vector

a standard represen tation

-

OP : (O,O)(u,v)

[see. 4-23

we

not the p o s i t i o n ,

shall assign to

a s the located v e c t o r from the origin to the p o i n t

as illustrated in Figure 4-2;

t h i s is the representation to which we shall

ordinarily refer unless otherwise s t a t e d .

Figure 4-2.

-

Representations of the column v e c t o r located vectors

OP

-e

and

-

Similarly, of course, the components o f the located vector considered as the entries of a row vector.

as

QR. A

can be

For the p r e s e n t , however, we shall

consider only column vectors and the corresponding geometric v e c t o r s ; in this chapter, the term "vector" will ordinarily be used to mean "coLumn vector,'' not "row vector" or T'locatedv e c t o r . "

The length o f t h e located vector is called the length or

Using the symbol

IlVlI

norm

-

OF,

to which we have previously referred,

of the column vector

to stand f o r the norm of

V,

we have

142 u

Thus, i f

and

v

are n o t both zero, t h e direction cosines of

u

and

I IVI 1

v I IVI I

are

'

respectively; these are also called the direction cosines of the column vector

v. The association between column or row v e c t o r s and d i r e c t e d

l i n e segments,

introduced in t h i s section, is as applicable to 34imensional space as it is to

The only difference is that: the components of a

the 2 d i m e n s i o n a l plane.

located v e c t o r in 3-dimensional space will be the entries of a column or row

v e c t o r of order 3 , not a column or row v e c t o r o f order 2 . In the r e s t o f this chapter and i n Chapter 5 , you will see how Theorems 4-1

and 4-2

can be i n t e r p r e t e d t h r o u g h geometric operations on locared vectors

and how the algebra of matrices leads t o beautiful geometric r e s u l t s .

Exercises 4-2 1.

Of the following pairs of v e c t o r s ,

which have the same l e n g t h ? 2.

Let

V = t

t=1,

In

.

which have the same d i r e c t i o n ?

Draw arrows from the origin representing

t = 2 ,

t=3,

t = - 1,

t = -

2,

and

each case, compute t h e Length and direction cosines of

V

t = - 3 .

V.

for

143 3.

In

a rectangular c o o r d i n a t e plane, draw t h e standard representation for

Use a d i f f e r e n t c o o r d i n a t e p l a n e

each of the following s e t s of v e c t o r s .

far each s e t of v e c t o r s .

Find the length and d i r e c t i o n c o s i n e s of each

vector :

(el

,I:[ [:I,

Draw the line segments

I:[

representing

(a)

rn = 1,

b = 0;

(b)

rn = 2,

b = 1;

(c)

m =

- 1/2,

+

and

V

if

[ i] . t = 0,

_+

1,

+ 2,

and

b = 3.

In e a c h case, v e r i f y that t h e corresponding set o f five p o i n t s

(x,y)

lies

on a line.

5.

n o

column vectors

are called p a r a l l e l provided their standard geometric

representations l i e on the same l i n e through the origin,

If

A

and

B

are nonzero parallel column vectors, de terrnine the two p o s s i b l e relations h i p s between the d i r e c t i o n c o s i n e s o f

A

and t h e direction cosines of

B. 6.

Determine a l l the vectors of the form

that are parallel to

4-3.

Geometric I n t e r p r e t a t i o n of the Multiplication of a Vector by a Number The geometrical significance of the multiplication of a vector by a number

is readily guessed on comparing t h e geometrical representations of the vectors V,

Zv,

and

-2V

for

By definition,

while

Thus, as you can see in Figures 4-3

V

and

2V

and 4 4 , the standard r e p r e s e n t a t i o n s of

have the same direction, while

p o i n t i n g in the opposite d i r e c t i o n .

F i g u r e 4--3.

The product o f

V

5, while those for 2V by

2

and

is represented by an arrow

The l e n g t h o f the arrow a s s o c i a t e d w i t h

V

Figure 4 4 . The p r o d u c t o f a v e c t o r and a negative number.

a vector and a p o s i t i v e number.

i s

-2V

-2V

each have length

10. Thus, m u l t i p l y i n g

produced a stretching o f the associated geometric vector to twice its

original Length while leaving i t s d i r e c t i o n unchanged.

Multiplication by

not only doubled the length of the arrow but a l s o reversed i t s d i r e c t i o n .

-2

These observations lead us to formulate the following theorem.

Theorem 4-3. and Let

r

be

a

Let the d i r e c t e d line segment

number.

Then the vector

rV

the r e p r e s e n t a t i ~ nof

Proof. -

Let

V

rV

G;

is opposite to that of

be the v e c t o r

Now,

hence,

This p r o v e s the f i r s t p a r t of the theorem.

the second part o f t h e theorem is certainly true.

the direction cosines of

-

PQ

are [sac.

4-33

V

is represented by a directed l i n e

8. If

having length Irl times the length of sentation o f rV has the same direction aa

represent the vector

r > 0 the repreif r < 0, the d i r e c t i o n of

G.

U

and

I IVI I

while those of

rV

ru

>

V

and

IIVII

'

are

Irl

If r

v -

= r,

I

0, we have

and

1IVlt

rv

Irl

l lVl l

'

whence i t follows that the arrows a s s o c i a t e d w i t h

have the same d i r e c t i o n cosine^ and, therefore, the same d i r e c t i o n .

rV

If r < 0, we have ated w i t h

Irl =

- r,

-

and the direction cosines of the arrow associ-

are the negatives of those of PQ.

rV

representation o f

rV

is o p p o s i t e t o that of

Thua, the d i r e c t i o n

G. This

o f the

completes the proof of

the theorem. One way of s t a t i n g part of the theorem j u s t proved

a number and

V

Exercise 4 - 2 4 } ;

i s a v e c t o r , then

V

and

is t o say that if

r

is

rV are p a r a l l e l vectors (see

thus they can be represented by arrows Lying on the same line

through the origin.

On the other hand, i f the arrows representing two v e c t o r s

are parallel, i t i s easy to show that you can always express one of the vectors as the product o f the other vector by a s u i t a b l y chosen number.

ing d i r e c t i o n cosines, i t is easy

to

Thus, by check-

v e r i f y that

are parallel vectors, and that

In the exercises that follow, you will be asked to show why the general r e s u l t

illustrated by this example holds true.

Exercises 4-3 1.

Let

L

be the s e t o f a l l vectors parallel t o the v e c t o r

rhe following blanks

so as to

[;I

.

Fill

produce in each case a true statement:

in

2.

(e)

f o r every real number

ty

(f)

f o r every real number

t,

[-"LC] L;

Verify graphically and prove algebraically that the vectors in each of the

In each case, express the first vector as

following pairs are parallel.

the p r o d u c t o f the second vector by a number:

3.

Let

V

parallel.

4.

5.

a nonzero vector such that

W

be a vector and

Prove that there exists a real number

Prove: (a)

If

rV=

(b)

If r v =

[:]

and

r

[i]

and

v i

V

Show that the vector

+ rV

1

4-4.

[i] ,

[:]

Y =

r

then

f rVlf

if

V

r <

= IlVll

- 1. I1

have two vectors

V

and

W,

[sec.

V =

b3j

are

r

> - 1, -

. 0 9

V

if

Show also that

+ rl.

Geometrical Interpretation o f the Addition of Two Vectors

If we

W

such t h a t

r

has the same d i r e c t i o n as

and the opposite direction to

IlV

then

0,

and

V

and

W =

i] ,

their sum

14a is, by d e f i n i t i o n ,

To i n t e r p r e t the addition geometrically, l e t us return momentarily to the conc e p t of a "free" vector.

Previously we have associated a c o l u m vector

w i t h some located vector

such t h a t

c = x2

- xl and d = y 2 - y 1.

In p a r t i c u l a r , we can associate W

with the vector

A:

We use

V

+W

A

to represent

W

(a,b)(a i - c ,

b +dl.

and use t h e standard representation f o r

in Figure 4 - 5 .

Vector addition.

Figure 4-5. A l r o , we can represent

V

as the located v e c t o r

0 : (c,d)(a

+ c,

[see.

b

4-41

+ d)

V

and

Ika and o b t a i n an alternative representation.

If the tw p o s s i b i l i t i e s are d r a m

on one s e t of coordinate axes, we have a parallelogram in which the diagonal

represents the sum; see Figure 4-6.

Figure 4-6.

Parallelogram rule for a d d i t i o n .

The parallelogram rule is often used in physics to obtain the resultant

when

two

forces are acting from a single p o i n t .

Let us consider now the sum of three vectors

We choose the three located vectors

-

PQ : ( a , b ) ( a and

to represent

5 V, W,

:

and

((a

+ u

C,

+ c, b

b

+ d)(a

+ d) -I- c

$.

e,

b f d

+-

respectively; s e e Figure 4--7.

f)

Figure 4-7.

The sum

+ W + U.

V

Order of a d d i t i o n does not a f f e c t the sum although i t does a f f e c t the geometric r e p r e s e n t a t i o n , as i n d i c a t e d i n Figure 4-43.

Figure 4 4 .

If V of

V

+W

and

W

The sum

V

+U

f

W.

are parallel, the construction of the proposed representative

is made i n the same manner.

Theorem 4-4. If the vectors

V

The d e t a i l s w i l l be l e f t to the s t u d e n t . and

W

[sec. 4-41

are represented by the directed

-

Ija

-L

rr,

line segments

Since

OP

and

V -W = V

V .t W

respectively, then

PQ,

+ (-W),

is represented by

OQ.

the operation of subtracting one vector from

another o f f e r s no essentially new geometric idea, once the construcrion o f

is undets t o o d representtng

.

It is useful to note, however, that since

the Length of t h e v e c t o r

P:

(u, v ) and

lllus trates the construction of the geome e r i c vector

Figure 4-9

V - W.

T: (r,

V

-W

equals the d i s t a n c e between the points

s).

T*

Figure 4-9.

The subtraction

/

of

T: (r, a )

vectors,

V

- W.

Exercises 4 4 1.

Determine graphically the sum and difference vectors.

+

of the following pairs of

Does order matter in cons tructfng the sum?

the difference?

152 2.

Illustrate graphically the associative law:

3.

Compute each of the following graphically:

4.

State

5.

the geometric significance of the following equations:

I:[

(b)

V + W + U P

(c)

V + W f U + T =

*

I:[

Complete the proof of b o t h parts a f Theorem 4-4.

4-55, The Inner Product of Two Vectors Thus f a r in our development, we have i n v e s t i g a t e d a geometrical interpret a t i o n for the algebra of vectore.

We have represented column vectors of order

2 as arrows in the plane, and have established a one--twne correspondence b e tween t h i s s e t of column vectors and the set: of d i r e c t e d l i n e segments from the

origin a f a coordinate plane.

The algebraic operationa o f a d d i t i o n of two

vectors and of multiplication of a vector by a number have acquired geometrical

s Fgni ficance

.

But we can a l e o reverse our point of view and see that the geometry of

vectors can lead us to the consideration o f a d d i t i o n a l algebraic structure.

For instance, if you look at the p a i r of arrows drawn in Figure 4--10, you may

c m e n t that they appear t o be mutually perpendicular.

You have begun t o

Figure 4-10.

Perpendicular vectors.

talk about the angle between the pair of arrows.

Since our vectors are located

vectors, the following d e f i n i t i o n is needed. Definition 4-2.

The angle between tm, vectors i s the angle between the

standard geometric representations of the vectore Let us suppose, in general, t h a t the points

and

R,

w i t h coordinates

8,

POR.

If

Consider the angle

in the triangle

We can compute the cosine of triangle

P, w i t h coordinates (a,b) ,

are the terminal points of two geometric

(c,d),

v e c t o r s with initial points a t the o r i g i n .

denote by the Greek letter

.

0

IOPI , IORI , and

WR

by applying the Law of cosines to the

lPRl

are the lengths o f the sides of

The angle between two vectors.

I-.

which we

of Figure 4-11.

the triangle, then by the law of cosines we have

Figure 4-11.

WR,

4-51

Thus,

= 2(ae f. bd)

.

Hence, IOPI IORI cos 61 = ac

+ bd.

(1l

The number on the right--hand side of this equation, although clearly a function

of the two vectors, has n o t heretofore appeared explicitly.

L e t us give it a

name and introduce, thereby, a new binary operation f o r vectors. Definition 4-3.

The inner product of the vectors

is the algebraic sum of the products of corresponding entries.

]

[;]

-

sc

Symbolically,

+ bd.

We can similarly define the inner product of cur row vectors: ac

+ bd.

[ a b] rn

[= d]

Another name for the fnner product of two vectors is the "dot product" of

the vectors. number. say

You n o t i c e

that the inner product of a pair of vectors is simply a

In Chapter 1, you met the product of

[ab]

times

[i] ,

and found that

a row vector by a column vector,

-

w the product being a

1 x 1 matrix.

As you can observe, these two k i n d s of

products are closely related; for, i f

[:]

and

[

,

vt = [a

w have

vt"

=

V

and

are the respective vector8

and

b]

[ae

W

-

+ bd]

[V

w] .

Later we shall exploit this close connection between the ttro product& i n order

ro deduce the algebraic properties of the inner product from the known properties of the matrix product. Using the notion of the inner product and the formula (1) obtained above, we can s t a t e another theorem.

We shall epeak

o f the

cosine of the angle included

between two column (or row) v e c t o r s , although we realize that we are actually

referring to an angle between aseociated directed l i n e segments. Theorem 4-5.

The inner product of two vectors equals the product o f the

lengths of the vectors by the c o s i n e o f t h e i r included angle.

V where

0

.

w

=

l lVl l

i s the angle between the veetore

Theorem 4--5 has been proved in the case

l l W l l cos 8 ,

V

W. in which V and

and

W

a r e not

If we agree t o take the measure of the angle between tul

parallel vectors. parallel vectors

Symbolically,

to

be '0

or

180'

according as the vectors have the same or

opposite directions, the result s t i l l holds.

Indeed, as you may recall, the law

of cosines on which the burden of the proof rests remains valid even when the three vertice~of the "triangle"

Collinear vectors in the same direction.

Figure 4-12.

POR are collinear ( ~ i g u r e s4-12 and 4-13).

Figure 4-13. Collinear vectors in o p p o s i t e direc t i o n e .

Iss The relationship

CoroLLary 4-3-1.

holds for every vector

V.

The corollary follows a t once from Theorem 4-5 by t a k i n g case

8 =

V =

]

,

we have

2

while

v m V = a 2 + b ,

Two column vectors

and

in which

To be sure, the result a l s o follows iwnediately from the facts

. ' 0

t h a t , f o r any vector

OP

V = W,

V

and

W

IIVII =

d m .

are said to be orthogonal i f the arroua

them are perpendicular to each o t h e r .

OR

is orthogonal to every vector.

the null vector

cos

90'

In particular,

Since

= cos 270'

= 0,

V

are orthogonal i f and only i f

we have the following result:

Corollary 4-+2.

The vectors

You will note that the condition

i f either We

V

or W

and

W

V a W =0

i s automatically s a t i s f i e d

i s the null vector.

have examined 8ome of the geometrical facets o f the inner product of two

vectore, b u t Let us now look a t same o f i t s algebraic properties.

Does i t sat-

i ~ f yconarmtative, associative, or other algebraic laws we have met i n studying

number aystems? We can show that the ctmrmurative Law h o l d s , that is,

For if V

and W

are any pairs of 2 x 1 matrices, a computation shows that

But V% =

W] , while

[V

W'V

-

[W

.

.

V]

Hence

It is equally poseible products.

Indeed, the products

To evaluate V

a (W

o f the vector

V

(W

U)

and

(V

W)

U

are meaningless.

W

U.

But the inner product is defined for

column vectors and not for a vector and a number.

two

V(W

a

U)

Inci-

should n o t be confused with the meaningless

The former product has meaning, for it is the product of the vector

by the number W

V

4

U) , for example, you are aaked t o find the inner product

dentally, the product

(W a U).

V

wlth the number

two row vectors or

V

t o show that the aaaociative Law cannot hold for inner

U.

In the exercises that follow, you will be asked t o coneider gome o f the In particular, you will be asked

other p o s s i b l e properties of the inner product.

ro establish the following theorem, the f i r s t part of which was proved above.

If V,

Theorem4-6.

W,

D are column vectors of order 2,

and

and

is a r e a l number, then (a)

V m W n W o V ,

(b)

(rV)

(d)

V a

V

(e)

if

VeV-0,

W

a

2 0;

r(V a W),

and

then V -

Exercises 4-5 1.

Compute t h e cosine

following pairs:

o f the angle between the two vector6 i n each of the

r

In which cases, if any, are the vectors orthogonal ?

In which

cases,

if any,

are the vectors parallel?

2.

Let

ELz[:]

and

Show that, for every nonzero v e c t o r

V

.

3.

(a)

Prove that

two vectors

[:I.

V, V • EZ

El and

1 IVI 1

are the direction cosines of

E2=

I IV1 I

V.

V

and

W

are parallel if and o n l y i f

Explain the significance of the s i g n of the right-hand side o f t h i s equation. (b)

Prove that

and write this inequality in terms of the entries of (c)

Show also t h a t

V

W

< -

V

and

W.

llVll IlWll.

4.

Show t h a t i f

5.

Pill in the bLanks i n the following statements s o a s to make the resulting

V

is the null v e c t o r then

sentences true: (a)

The vectors

[:] I'!-[ and

are parallel.

(b)

The v e c t ~ r s

[:I I : [ and

are orthogonal.

The vectors

and

are

The vectors

and

are para1 le 1.

For every p o s i t i v e real number

(e)

[ (f)

and

[]

end

the vectors

are orthogonal.

For e v e r y negative real number

[:I

t,

[:]

t,

the vectors

.

are orthogonal

6. Verify that parts (a) - (d) of Theorem 4-6 are true if

7.

Prove Theorem 4 - 4 (a)

by using the definition of the inner product of two vectors;

(b)

hy using the fact t h a t the matrix product

V%

s a t i s f i e s the

equation

8.

+2

Show t h a t , in each of the following s e t s of vectors,

V

I IV

+ Wl l 2

every pair of vectors

9.

-

= l lVl iZ

Pmve t h a t

orthogonal,

V

and

V

(V

+ W)

(V

+ W)

and W.

T are p a r a l l e l , and

Do t h e same relationships hold for the s e t

T

and

W

V

.

W

and

+ W

l lUl l Z for

are

are orthogonal:

160 10. Let

V

T and 11.

b e a nonzero v e c t o r .

V

are parallel.

Suppose that

Show that

W

12. Let

V

=

[z] ,

[-:I

and

r,

Show that the v e c t o r s

14.

Show that i f

A =

V

[:]

and

I IAI i Z

15.

Show that the vectors

16.

Show that the equation

holds f o r a l l vectors

17.

I

[i] ,

B =

I B Il 2

- (A

V

and

W

V

and

W.

t

holds for all v e c t o r s

V

+ WII

and

W.

the v e c t o r s

Show t h a t if

such that

8)

2

then

= (ad

- bc) 2 .

are orthogonal if and o n l y i f

Show t h a t the inequality IIV

t,

are orthogonal if and only if

W

and

are orthogonal, while

are orthogonal.

are orthogonal, there exfs ts a real number

13.

V

and

a,

is n o t the zero v e c t o r .

V

where

and

T are then orthogonal,

and

Show t h a t , f o r every s e t of real numbers

[i]

W

< -

IIVll

+

IlWll

W and

V

4-6.

Geometric Considerations

In S e c t i o n That is, if

4 4 , we saw that two parallel v e c t o r s determine a parallelogram.

.A

u

I:[

and

8 =

[i]

are t w o nonparallel vectors w i t h i n i t i a l p o i n t s a t the origin, then the p o i n t s

Pt(a,b),

0:(0,0),

R;(c,d)

and

S:(a + c ,

parallelogram ( F i g u r e 4-14.)

determine the area of the parallelogram

R:

b +d)

are the vertices of a

A reasonable question to ask is: '!How

(c,

ME?"

d)

A parallelogram determined by

Figure 4-14,

can we

vectors.

As you recall, the area of a parallelogram equals the product of the lengths Thus, in Figure 4--15, the area of the parallel*

of its base and its a l t i t u d e .

gram

is

of the altitude

blh,

where

bl

is the length of s i d e

NM and h

i s the length

KD.

Figure 4 1 5 .

Determination of the area of a parallelogram. [see, 4-61

162 ~ u ti f

NKL

angle

b2

is the length of side

or a n g l e

NK,

we have

KNM,

blb2 l s i n 8 1

Hence, the area o f the parallelogram equals Returning to Figure 4-14 and letting A

and

B,

is the measure of e i t h e r

8

and

we can now say t h a t if

G

2

.

be the angle between t h e vectors

0

1s the area of parallelogram

G

= I t*,i2

I IBI I

2

PORS,

2

sin 8 .

Now sin

2

= 1

- cos2

e.

It follows from Theorem 4-5 that

COS

2

e

(A

=

.

B)

r:

I IA1 l 2 I I B I I

2

'

therefore,

Thus, we have

G~

= 1 1 ~ 1 1 1~ ~ 1 1 ~(A.

B)

2

.

It follows from the result of Exercise 14 o f the preceding section that 'G

= (ad

- bc) 2 .

Therefore,

G = lad

- bcl.

then

E

But ~trix

f

theorem,

ad

- bc

[ E] . L"

is the value of the determinant 6(D), where D I s the For easy reference, l e t ue vrtte our r e s u l t in the form of a

"1

Theareri 4-7.

The area o f the parallelogram determined by the standard

;I.

representation of the vectors

D =

[;

Corollary 4-7-1. if S(D) =

equals

The vectors

16(D)I,

where

are parallel if an? only

0.

The argument proving the corollary is l e f t

at3

an exercise f o r the s t u d e n t .

You n o t i c e that we have been l e d t o the determinant of a examining a geometrical interpretation o f vectors.

2 x 2

matrix in

The role of matrices in t h i s

interpretation will be further investigated in Chapter 5.

From geometric considerations, you know that the cosine of an angle cannot exceed 1 in absolute value,

and that the length of any side o f a triangle cannot exceed the sum of the lengths of the other two sides,

+ PQ.

OQ 5 OF

Accordingly, by the geometric interprets t i o n of column v e c t o r s , for any

V

- [;]

and

Y

- [;]

we must have

i

and

IlV

+ Wll

5 lIV11 + [see.

4-61

IlWlI.

164 But can these i n e q u a l i t i e s be established algebraically?

L e t us s e e .

The i n e q u a l i t y (1) is equivalent to

(V

. . . w)~<

(V

V)(W

W),

t h a t is,

or

- as

we see when we m u l t i p l y our and simplify

- to

But t h i s can be written as

0

5 (ad

- bc) 2 ,

which certainly is v a l i d s i n c e the square of any real number is nonnegative. Since

ad

- bc

= 6(D),

where

you can see that the foregoing result is consistent w i t h C o r o l l a r y 4-7-1,

that i a , the sign of equality holds in (I) if and only i f the vectors W

are p a r a l l e l . As for the inequality ( 2 ) , i t can be written as

which simplifies t o

which a g a i n is valid since 0

< - (ad

- bc)".

Is-.

b63

above;

V and

165 This rime, the s i g n of equality holds if and only if the vectors

V

and

w

are parallel and

that is, i f and only i f the vectors

V

and

W

are parallel and in the same

direction.

If you would like t o look further i n t o the study of inequalitiee and their applications, you might consult the SMSG Monograph

"An Introduction t o

Inequalities," by E. F. Beckenbaeh and R. hllman.

1.

Let

-

OP

Exercises 4 4 repreeent the vector

A,

end

6?

the vector

if

area of triangle TOP

2.

Compute the area o f the triangle w i t h vertices:

3

Verify the inequalities (V.

and

for the vectors

wl2

< (V.

V)(Y.")

B.

Determine the

4-7.

(a)

V = (3,4)

and

W = (5,121,

(b)

V = (2.1)

and

W

=

(c)

V = (-2,-1)

and

W

= (4,Z).

(4,2),

Vector Spaces and Subapacea

Thus f a r our discussion of vectors has been concerned essentially u i t h individual vectors and o p e r a t i o n s on them.

In t h i s section we shall take a

broader point of view.

It will be convenient

t o have a symbol for the s e t of

2 x 1

matrices.

Thus we let

where

R

L s the s e t o f real numbers.

The s e t

H

together u i t h the operations

of addition of vectors and of multiplication of a vector by a real number is an example of an algebraic system called a vector space.

D e f i n i t i o n 4-4.

A s e t of elements is a vector space over the s e t

R

of

real numbers provided the following conditions are a a t i s f i e d :

( a ) The sum of any tw elements of the set is aLao an element of the set.

(b)

The product of any element of the s e t by a real number is also an element of the set.

(c)

The Laws I and If o f Theorem 4-2 h o l d .

In applying laws I and I f , space,

0 will denote the zero element of the

vector

Let us emphasize, houever,that: the elements of a vector space are n o t

necessarily vectors i n the sense thus far discussed in this chapter; f o r example, the s e t of

2 x 2

matrices, together w i t h ordinary m a t r i x addition and

multiplication by a number, forms e vector space.

Since a vector space consists of a s e t together with the operations of

addition of elements and of multfplication o f elements by real numbers, strictly speaking ue should n o t use the same symbol for the set o f elements and for the vector space.

But the practice i s not likely t o cauae confusion and will be

[sea.

4-63

167 £0 1 lowed.

A completely trivial example of a vector space over

sisting of the zero vector set

It

of vectors parallel

is

alone.

Another v e c t o r space over

that is, t h e

to

is the set con-

R

is the

R

set

evident t h a t we are concerned uith subsets o f

H

in these two examples.

Actually, these subsets are subspaces, in accordance with the following de finition.

D e f i n i t i o n 4--5, Any nonempty subset

F

of

H

H

is a subspace of

provided the following conditions are s a t i s f i e d :

The sum o f any two elements o f

is a l s o an element of

F

by a real number i s an element o f

F

The product o f any element o f

F.

By definition, a subspace must contain a t least one element

rV f o r real numbers r . therefore has the zero vector as an element,since f o r r

m u s t contain each

H

V

F,

and also

Every subspaee a f

of the produces

=

0 we have

It is eeay to see that the set consisting of the zero vector alone is a subspace. We can a l s o verify that the s e t of a l l vectors p a r a l l e l t o any given nonzero

vector is a subspace.

Other than

H

i t s e l f , subsets o f these two types are

the only subspaces of H. Theorem 4 - 8 .

Every subspace of

H consists of e x a c t l y one o f the folio*

ing: the zero vector; the set of vector^ p a r a l l e l to a nonzero vector; space

H

itself,

Proof.

If

P

is a subspace containing only one vector, then

since the zero v e c t o r belongs to every subs pace.

[see,

b73

the

168 contains a nonzero vector

F

ff

f o r real

V,

Accordingly, if a l l vectors of

r.

F

then

F

contains all vectors

rV

are parallel t o

V,

i t

V,

F

is actually

follows

that

If F also contains equal to

a vector

W n o t parallel t o

then

H, as we shall now prove.

Let

be nonparallel vectors in the subspace

be any other v e c t o r of

H.

F,

and l e t

We shall show that:

Z

is a member o f

F.

By the d e f i n i t i o n of subspace, t h i n will be the case if there are numbers x

and

such that

y

that is,

These can be found i f we can solve the system

for

x

since

ad - bc

and V

and

+O

in terms of the known numbera,

y

W

a , b, e , d ,

r,

and

s.

are n o t parallel, i t follows (see Corollary q - 1 ) that

and therefore the equations have the solution

But

169 Since

is a subspace that contains

F

their sun

Thus every vector

2.

a subset of

F.

F

But

2

and

V

of

H

it contains

W,

must belong to

xV,

yW,

F ; that is,

H. Accordingly,

is given to be a subset of

and

H F

=

is

H.

Using the ideas o f Section 4 4 , we can g i v e a geometric i n t e r p r e t a t i o n t o Equation (1) .

L e t the nonparallel v e c t o r s

standard representations

i%

OT.

be represented by t o one of them must

through

T

-OP

and

4

Since

V

and

W

in the subspace

O?i,

respectively; s e e Figure 4-16,

and

O?R

5

and

d,

have

Let

2

are n o t parallel, any line parallel

intersect the line containing the other.

p a r a l l e l to

F

and let

S

which these l i n e s i n t e r s e c t rhe lines containing

-

and

OR

(1

and

Draw the l i n e s

-

be t h e p o i n t s in

OP

respectively.

Then

Figure 4-16. Representation o f an arbitrary vector Z a s a linear combination of a given pair of nonparallel v e c t o r s V and W.

But x

is parallel to and

y

6$

to

OR.

Therefore, there are real numbers

such that

6?j Hence,

4

and

=

ZP and

d

4

OS = yOR.

This ends our discussion

of

Theorem 4-7 and introduces the important concept

o f a linear combination:

If

Definitfon 4--6. where

x

and

y

a vector

can be expressed in the form xV

Z

are real numbers and

caLled a linear combination

V

of

and

V

and

W

are vectors, then

+ yW, 2

is

W.

Further, we have incidentally established the useful f a c t s s t a t e d in the following theorems:

Theorem 4-9. pair of vectors in

Theorem 4-10.

A subspace

F

contains every linear combination of each

F. Each vector of

H can be expressed as a linear combination

of any given pair of nonparallel vectors in

H.

For example, t o express

as a linear combination of

we must determine

real numbers

x

and

y

such that

Thus, we must solve the s e t of equations

We f i n d the unique s o l u t i o n

x = 2

and

y = 1;

that is, we have

X f you observe rher the given vectors W = 0

V

are orthogonal, that is,

V

(see Corollary 4-5-2

second method of s o l u t i o n may occur ro you.

then for the products

V

2

V =

Z

and

allVll

in the foregoing example

W

and

Z

For if

you have

W

2

on page 156) then a

and

W = bltWll

Z

2

.

But

V a 50, Z

Z

W = 25, 1 1 ~ 1 =1 ~25, and

-

IIW112

25.

Hence, 50 = 25a and

,

25 = 2%

and accordingly a = 2

It

bSl.

and

i s worth noting t h a t the representation of a v e c t o r

combination of two given nonparallel vectors

Z

as a Linear

unique; that i s , if the vectors

is

V

and W

y

can be chosen in exactly one way (Exercise 4--7-11, below) so that

are not p a r a l l e l , then for each vector

The pair o f nonparallel vectors real numbers basis

.

x and y

V

and

W

the c o e f f i c i e n t a

Z

is called a basis for

are called the coordinates of

\ li] r

In the example above, the vector

Z

x

and

H, while the

r e l a t i v e to that

C l

has coordinates

2

and

1

reLative to the basis

In particular, the pair of vectors natural basis for point

(u,v)

H.

and

[y ]

1s called the

This basis allows us to employ the coordinates of the

associated with the vector

t o the basis; thus,

[i] V =

[:I

as the coordinates relative

Since every vector of

H

can

be expressed as a linear combination of any

pair of basic vectors, t h e basis vectors are said to span the vector space.

The minimal number of v e c t o r s t h a t span s vector space is c a l l e d the dimension of the particular space. For example, the dimension of t h e vector space H sense, the s e t

is

2.

In the same

F

is a subspace of dimension f o r this subspace.

Note that neither

1.

I:[

nor

[;I

is a baais

(What is?)

In a 2-space, t h a t is, a vector space o f dimension 2 , it i s necessary that any s e t of b a s i s vectors be linearly independent, Two vectors

D e f i n i t i o n 4-7.

only i f , f o r a l l real numbers

x = y = 0.

implies

Otherwise,

and

W

V

are linearly dependent.

w h i c h indicates that

and

and

y,

and

are linearly independent f f and

W

the equation

W

are said to be l i n e a r l y dependent.

V =

For example, let

V

x

V

V

and

W

Note that

are parallel,

Exercise 4-7 1

i

1.

Express each o f the following v e c t o r s as linear combinations of

,

i

1

2.

and i l l u s t r a t e your answer g r a p h i c a l l y :

In parts

[-:I

[:I

and

through ( i ) of Exercise 1 , determine the coordinates o f each

(a)

of the vectors relative to the basis

and

[:].

3.

Prove t h a t the following s e t i s a subspace of

H:

.

Prove that, for any given vector

[rW : r E R]

W,

the s e t

is a subspace

of H. 5.

I

(a)

I-t

I

- I:[

,

determine which o f the following subsets of

I

I6

For

all V

with

u = 0,

(b) a l l (c)

6.

Prove that

Show that

(d)

all

V

with

V with v to an integer,

equal

(e)

all

V

with

all

rational,

(f)

all

V

with

F

V

with

u

is a subspace o f

i f and only if

H

linear cmnbination of two vectors in 7.

H are subspacee:

F

2u

- v = 0,

u + v = 2, uv = 0.

contains every

F.

cannot be expressed as a l i n e a r combination of the

vectors

! 8.

t

Describe the s e t of a l l linear combinations o f two given parallel vectors,

[aec.

4-73

9,

F1

Let

F2 be subspaces of

and

In proving Theorem 4-10,

Prove t h a t the s e t

F of all

F 1 and F2 is a l s o a subspace.

v e c t o r s belonging to both

10.

H.

we showed that:

vectors, then each vector of

if

V

and

W

are n o t parallel

H can be expressed as a linear combination

If each vector of H has a representation as a linear combination of V and W, then V and W are not

of

V

and

W.

Prove the converse:

parallel.

11.

Prove that i f vector a

12.

and

V

and

in the form

b

W

are not parallel, then the representation o f any

aV

+ bW

i s

unique;

that i s , the coefficients

can b e chosen in exactly one way.

Show that any vector

can be expressed uniquely as a linear combination of the basis vectors

4-8.

Summary

In this chapter, we have developed a geometrical representation - namely, d i r e c t e d line segments

- for

2 x 1 matrices, or coluum vectors.

Guided by

the definition o f the algebraic operation o f addition o f v e c t o r s , we have found

the "parallelogram law of additiont' of directed Line segments.

The multiplica-

t i o n of a vector by a number has been represented by the expansion or contraction o f the corresponding directed line segment by a factor equal t o the number, w i t h

the sign of the factor determining whether or not the direction of the Line

segment is reversed.

Thus, from a set of algebraic elements we have produced a

s e t of geometric elements.

Geometrical observations in turn led us back to

additional algebraic concepts.

Also in this chapter, we have introduced the important concepts of a vector space and l i n e a r independence.

Since the nature of the elements of a vector space i s not limited except by the postulates, the vector space does not necessarily c o n s i s t of a set whose

175 elements are

2 x 1

column v e c t o r s ; thus i t s elements might be

n x

matrices,

n

real numbers, and so on.

P of linear and constant polynomials

For example l e t us look at the s e t

with real coefficients, that is, the s e t

under ordinary a d d i t i o n and multiplication by a number.

The sum of two such

polynomials,

is an element oi t h e s e t s i n c e the sums

The product of any element

of

P

+ a2

and

bl

+ b2

are real numbers.

P by a real number, c(ax

is also a member of

a1

+ b)

= acx

since the products

+ bc, ac

and

bc

are real numbers.

We

can similarly show that addition is c o m u t a t i v e and as~ociative,t h a t there is

an i d e n t i t y for addition, and that each element has an additive inverse; thus, the laws 5: o f Theorem 4 . 2 are v a l i d .

In l i k e f a s h i o n , we can demonstrate that

laws I1 are s a t i s f i e d : both d i s t r i b u t i v e l a w s hold; the mu1 tiplication of an element by two real numbers is associative; the product of any element

the real number

1

is

p

itself; the product of

0 and any element:

by

p p

is the

zero element; and the product of any real number and the zero element is the

zero element. We have outlined the p r o o f t h a t the s e t

nomials is a vector space.

P

of linear and constant poly-

Thus the expression, "the vector,

meaningful when w e are speaking of the vector space

ax

+ by"

is

P.

The mathematics t o which our algebra has led us forms the beginnings of a

discipline called "vector analysis," which i s an important t o o l in classical and modern p h y s i c s , as we11 as in geometry.

The "free" vectors t h a t you meet

in physics, namely, f n r c e s , velocities, etc., can be represented by our geometric vectors.

The study i n which we are engaged i s consequently of v i t a l importance

f o r physicists, engineers, and other a p p l i e d scientists, as well as for math-

maticians

.

Chapter 5

TRANSFORMATIONS OF THE PLANE Functions and Geometric Transformations

5-1.

You have discovered that one o f the most fundamental concepts i n your study

of mathematics is the n o t i o n of

a function.

appears i n the idea of a transformation.

In geometry the function concept

11: is the a i m o f this chapter t o recall

what we mean by a function, to define geometric transformation, and to explore t h e role of matrices in the study of a significant c l a s s of these transformations. You r e c a l l that a function from s e t

mapping from the elements of t h e s e t

element of A

A

B

to set

is

a correspondence or

there is associated exactly one element of

the domain of the function and the subset of

-

B

t o those ofathe set

A

B

The s e t

0.

onto which

such that with each

A

A

is

is mapped is the

In your previous work, the functions you net generally

range of the function.

had s e t s o f real numbers both for domain and f o r range.

Thus the function

s p b o l i z e d i n the form

is likely to be interpreted as a s s o c i a t i n g the nonnegative real number

the r e a l number

x.

xZ

v i th

Here you have a simple example o f a "real function'' of a

"real v a r i a b l e .I1 In Chapter 4 , however, you m e t a function domain the vector space

t lVl

V

H, and f o r its range the

having f o r its

s e t of nonnegative

real

numbers. In the present chapter, we shall c o n s i d e r functions rhar have their range as well as their domain in

H.

SpecificeLly,

we want to

find a geometric in-

terpretation for these "vector functions" of a "vector variable"; this is a

-

continuation of the discussion started in Chapter 3. considered i n their standard representations

OP,

A l l v e c t o r s will now be

so that they w i l l be p a r a l l e l

if and only if represented by c o l l i n e a r geometric v e c t o r s . Such a vector function will associate, with the point (x,y),

a point

geometric vector

-

P'

OP

with coordinate&

x

y

d

onto the geometric vector

OP'

.

Pf

of t h i s plane.

having coordinates

The function can, t h e r e

fore, be viewed as a process that associates with each point same point

P

Or we may say that it maps the

P of the plane

W e shall c a l l this process a transformation of

the plane into i t s e l f or a geometric transformation.

As a matter of f a c t ,

these transformations are of ten called "point transformations" i n conrras t t o

more general mappings i n which a p o i n t may be carried into a l i n e , a c i r c l e , or

For us, a geometric transformation i s a

some other geometric c o n f i g u r a t i o n .

helpful means o f visualizing a vector function of a vector variable.

As a

matter of c o n v e n i e n t terminology, we s h a l l call the vector that such a f u n c t i o n a s s o c i a t e s with

a given

vector

the image of

V

V;

furthermore, we s h a l l say

9 V ontD its image.

that the function

Let us look a t t h e simple function

This function maps each vector as

V,

V

onto the vector that has the same d i r e c t i o n

V.

but: that is twice as long as

Another way of asserting t h i s i s to

say that the function associates with each p o i n t

such that

P and

see Figure 5-1.

P'

P

of the plane a point

Pt

l i e on the same ray from the origin, but

You may therefore think

o f the function

uniformly stretching the plane by a factor

i n this example as

in a l l directions from t h e origin.

2

(Under this mapping, what is the point onto which t h e origin i s mapped?) As a second example, consider the function

This time, each

vector i s mapped onto

the vector having length equal and direction

opposite to that o f the given vector.

function associates with any point:

P

Viewed as a point transformation, the its "reflection" i n the o r i g i n ; see

Figure 5-2.

The function

combines both of the effects associated with

that of

V.

V

of

the preceding functions, so that the v e c t o r

is twice as Long as

V,

b u t has the o p p o s i t e direction to

Figure 5-1.

Figure 5 2 . The rransformation V -9-V.

The trans-

V

formation

-+ 2V.

Now, let us look a t the function

As in our first example, each vector is mapped by the function onto a vector

having the same d i r e c t i o n as the given v e c t o r . 1

is its own image.

greater than that of of

V

itself.

having length

But if

V,

> 1,

V

then the image of

is mapped onto

which is twice as long.

whose length i s 13,

V

has a length

with the expansion factor increasing with the length

Thus, the vector

2,

Indeed, every vector of length

The vector

has the image

[sec. 51)

w i t h length 1 6 9 .

On t h e other hand, for nonzero vectors of length less than

1, we obtain image vectors of shorter length, t h e contraction factor decreasing w i t h decreacing length of the original v e c t o r .

Thus,

is mapped onto

the Lmaze being half as long as the given v e c t o r .

Again, the vector

t h e length of the first vector being 5/7, while the length of i t s image i s 2 only ( 5 / 7 ) , or 2 5 / 4 9 . Although we may try to think o f t h i s mapping as a kind of stretching o f the p l a n e i n a l l d i r e c t i o n s from the o r i g i n , so that any point and its image are collinear with the o r i g i n , t h i s mental picture has also

to take into account the fact that the amount of expansion varies w i t h the

distance o f a given point from the origin, and that f o r p o i n t s within the circle of radius

about the origin the s e c a l l e d stretching is actually a compression.

1

We have been considering transformations of individual vectors; l e t us look a t c e r t a i n transformations o f the square

[:]

and

[

.

maps

r e s p e c t i v e l y , onto

As shown in Figure 5-3,

ORST, determined by the basis vectors

the f u n c t i o n

Figure 5-3.

Reflection in the o r i g i n .

Another transfontation that is readily visualized is the r e f l e c t i o n in the x

axis,

For t h i s mapping, the point

(x,y)

goes into the p o i n t

see Figure 5-4.

Figure 5 4 . That

is,

Reflection in the

x

the map is given by

Using a matrix, we may rewrite t h i s r e s u l t in the form [see.

5-11

axis.

(x,--y) ;

as

is e a s i l y v e r i f i e d . the square

NOW this unchanged; thus the vector

(1,O)

[]

y

i s

ORST,

leaves the point

mapped o n t o i t s e l f

.

The v e c t o r

is mapped o n t o Reflection in the

1 8 0 ~ in space about the

axis, or a rotation of

y

a x i s , can be expressed similarly:

as shown in Figure S 5 .

F i g u r e 5-5.

Rotation of

about the axes.

189'

Casual observation (see Figures 5-5 and 5-6) a

180'

the

r o t a t i o n about the

(x,y)

onto

axis and

90'

pLane are equivalent; they are n o t .

leaves t h e point (0,l)

y

(0,l)

-10).

may l e a d you to assume t h a t

rotation about the origin in

The f i r s t transformation

unchanged, whereas the second transformation maps

As a vector function, the

90'

rotation with respect t o

the origin is expressed by

Figure S 6 .

Rotation of

The transformations of Figures 5-2 size n o r the shape of the square.

does a l t e r size.

90

0

about the origin.

through 5 4 have altered neither the

The "stretching" function of Figure 5-1,

A "shear" that moves each p o i n t parallel t o the

x

axis

through a distance equel to twice the ordinate o f the p o i n t a l t e r s shape.

Con-

sider the transformation

which maps the basis vectors onto

The result is Figure 5-7.

a sheering t h a t transforms the

square into a parallelogrflm; see

What does the s t r e t c h i n g do t o shape?

f sec. 5-11

The shear to size?

Figure 5-7.

Shearing.

Another type of transformation involves a displacement or "translation" in the direction of a fixed vector.

V

The mapping

+ V + U,

where

U =

I:1

*

can be written in the form

One way of visualizing this function is to regard i t as translating the plane in

the direction of the vector

U

through a distance equal to t h e length of

U.

Tran~formationof two d i f f e r e n t t y p e s can be combFned in one function.

For

i n s tance , the mapping

V

1 +?(V + U),

where

U =

involves a translation and then a compression.

When the function is expressed

in the form

ue recognize more easily that every point P is mapped onto the midpoint the line segment joining P to the point: (3,Z); see Figure 5 8 . [sea. 5-11

of

Figure 5-8. The transformation V

Under t h i s mapping, the square

1 +-i.(V + U) .

will likewise be translated 1 toward the p o i n t (3,2) and then compressed by a factor , as shown in 2 Figure 5-9. This f i g u r e enables us to see that the points 0, R, S, and T ORST

-

are mapped onto midpoints of l f nes connecting these p o i n t s t o ( 3 , 2 ) .

Figure 5 - 9 .

Translation and compression.

A l l the vector functions d i s c u s s e d above map d i s t i n c t points of the plane onto d i s t i n c t p o i n t s .

This i s not always the c a s e ; we can c e r t a i n l y produce

functions that do not have t h i s property.

[sec.

Thug, the f u n c t i o n

~ l l

On the other hand, the trans-

maps every poinr of the plane onto the o r i g i n .

formation

maps the poinr

component as

the p o i n t

(x,y)

onto the point o f the

axis that has the same f i r s t

x

For example, every p o i n t o f the line

V

(3,O).

Since the image o f each point

a p e r p e n d i c u l a r Line £rum P carried or projected on the

t o the x

x

= 3

is mapped onto

can be located by drawing

P

axis, we may think of P as being

x

axis by a l i n e perpendicular t o this axis.

Con-

sequently, t h i s mapping may be described a6 a p e r p e n d i c u l a r or orthogonal projection o f the plane on the map

H

o n t o subspaces

of

axis.

x

You notice that these Last rw functions

H.

Since we have met examplea o f transformations t h a t map d i s t i n c t p o i n t s onto distinct points and have also seen transformations under which distinct points

may have the same image, i t i s u s e f u l to define a new term t o d i s t i n g u i s h be-

tueen these two kinds D e f i n i t i o n 5-1.

of

vector f u n c t i o n s .

A transformation frum the s e t

H onto the s e t

H

is one-

tcrone provided that the images of d i s t i n c t vectors are also d i s t i n c r v e c t o r s .

f

Thus, if the image of

V

is a f u n c t i o n from

H onto H

under the transformation

formulated symbolically as follows:

f,

and if we write

then Definition 5--1

The function

tion of H provided that, for vectors

V

and

U

f

Is a on-to-one

in H,

imp lies

Exercises 5-1 1.

Find the image of the vector

V

under the mapping

f(V)

for

can be transforma-

V

(1)

AV.

The study of the solution of systems of Linear equations thus leads t o the consideration of the s p e c i a l class of transformations on

in the form (11, where

A

is any

2 x 2

H

that are expressible

matrix with real entries.

These matrix

transformations constitute a very important class of mappings, having extensive

applications in mathematics, statistics, phyefcs, operations research, and

enatneering. h important properry of matrix transformations Is

chat: they are linear

mappings ; that is, they preserve vector sums and the products o f vectors with

real numbers. L e t us formulate these ideas explicitly.

Definition +2.

H

is a functfon

U

in H, we have

f

from H

H such that

into

(a)

for every p a i r of vectors

(b)

f o r every real number

Theorem 5-1.

Proof.

where

and

A linear transformation on

U,

A

Let

r

V

and

and every vector

V

in H,

we

have

Every matrix transformation is Linear.

f

be rhe transformation

is any real matrix of order 2 .

We must show that for any vectors

V

we have

A(V

+ U)

+ AU;

= AV

further, we must show that for any vector

V

and any real number

r we have

19 But these equali t i e g hold in virtue o f parts I11 (a) and 111 ( £ 1 o f Theorem 4-2 (see page 1 3 4 ) .

The l i n e a r i t y p r o p e r t y of matrix transformations can be used t o derive the following result concerning transformations of the subspaces of

A matrix

Theorem 5-2.

F'

maps every subspace

A

F

of

H.

H onto a subspace

H.

of

Ft

Let

Proof.

F1

To prove that

denote the s e t of vectore

i s a subspace of

we must show that the following state-

H,

ments are true:

For any pair of vectors

(a)

in

F'

PI

If and

Q

F',

the sum

P' + Q '

is

P'

F1 and any real number r,

in

rP'

is in

. Q'

and

in

F1.

For any v e c t o r

(b)

PI, Q'

are in

F',

then they must be the images of vectors

P

F; that is,

in

P'

=

AP,

Q' = AQ.

It follows t h a t P' end P

PI

+Q

f

Thus,

rP'

= AP + A Q = A ( P 4- Q),

is the image of the vector

Q'

is in

and hence

+ Q'

F?)

rP'

Hence,

(P'

f

i s the image of

Q') E F'.

rP.

+Q

in

F.

(Can

you t e l l why

Similarly,

But

is t h e image o f a vector in

Corollary 5-2-1.

P

rP

F;

f

F because

therefore,

Every matrix maps the plane

Isec. 5-21

rP'

F

is a subspace. E

F'.

H o n t o a subspace of H,

192 either the origin, or a straight Line through the origin, or H

itself.

For example, t o determine the subspaces onto which

H

(b)

itself,

we proceed as f o l l o w s .

For (a),

the vectors of

F

are o f the form

Hence,

F is mapped onto F',

Thus,

In other words,

A

[t]

the s e t of vector8 collinear with

;

t h t is,

maps the line passing through the o r i g i n w i t h slope -3

onto

the lfne through the origin with slope 1/2. A8

we

regards (b), we note that for any vector

have

Since

2x

+y

assumes a11 real values

numbers, it folloca that

H

as x

and

fs also mapped onto

entire plane onto the line [sec. 5-21

y

run over the set of real

F 1 ; that is,

A

maps the

Exercises L 2

I.

Let

A =

[i

:].

For each of the following values of the vector

V,

determine:

( i ) the v e c t o r into which A maps (ii) 2.

the line onto which

A

V,

maps the line containing V,

A certain matrix maps

I:[

into

[:] ,

and

[ i]

into

[r] .

Using t h i s i n f o r m a t i o n , determine the vector i n t o which the matrix maps

each of the following:

3.

Consider the f o l l o w i n g subspaces of

F3

=[:I

: Y E -

2x

I

,

H:

F4=H

itself.

194 Determine the subepaces onto which

F1, F 2 , F 3 , and

F4

are mapped by each

of the following matrices:

a

5.

=

[ ]

,

AV

(a)

Calculate

(b)

Find the vector

(8)

3

- [ '1 -2O

3

'

(GI

*g5

(dl

BA-

for

V

for which

Determine which of the following transformations of

H are linear, and

j u s t i f y your answer:

(e)

,V

71 (V

f

U), where

U =

6.

Prove t h a t the matrix A maps the plane onto the origin if and only if

7.

Prove t h a t the matrix only if

8.

Prove that

A

maps every vector of the plane onto itself if and

w5 maps the line

y = 0

IS any point

onto itself.

of

that line mapped onto

itself by t h i s matrix? 9.

(a)

Show that each of the matrices

H onto the x axis.

maps

(b)

Determine the s e t of aLL matrices that: map

(Hint:

H onto the x

You muat determine all p o s s i b l e matrices

corresponding t o each

V G H

A

axis,

such t h a t

there is a real number

r

for

which

[;I

In part cular, (1) mus

Y

[

andby

hold for s u i t a b l e

Determine the s e t of a l l matrices that map

11.

(a)

f o r all

(b)

V

is replaced

h )

LO.

Determine the matrix

r when

H onto the

y

axis.

such t h a t

A

V.

The mapping

multipliee the lengths of all vectors without changing their directions.

It amount6 t o a change of scale.

scale factor or scalar.

The number

Find the matrix

A

a

i~ accordingly called a

that yields only a change o f

scale:

12.

Prove that for every matrix

A

the s e t

[see. ?2]

F

o f all vectors

U

f o r which

196 . 13.

This subspace is called the kernel of the mapping.

H.

is a subspace of

Prove that a transformation

r

5-3.

and

U

into itself is linear if and only i f

of

H

and every pair of real numbers

s.

Linear Trans formations I n the preceding s e c t i o n ,

H

transformation of

formation of

H

we proved chat every matrix represents a linear

into 8. We now prove the converse:

into H

can be represented by a matrix.

Let

f be a linear transformation of H

Theorem 5-3.

relative to any given basis for

such t h a t , f o r all

Proof. ing

and

V

for every pair of vectors

of B

f

V

E

H,

into

H.

Then,

there e x i s t s one and only one matrix

A

H,

We prove f i r s t that there cannot be more than one matrix represent-

Suppose that there are t w o matrices

f.

Every linear trans-

A

and

B

such that, for all

V E kt, AV = f (V)

and

BV = E(V)

.

Then AV

for each

V.

that

A

A

-B

-B

=

f (V)

- f (V)

Hence, (A

Thus,

- BV

- B)V =

[i]

for all

maps every vector onto the o r i g i n .

is the zero matrix; therefore,

V t

X.

It follows ( E x e r c i s e 5 - 2 4 )

197 Hence, there is at most one matrix representation of

f.

Next, we show how t o f i n d the matrix representation for the linear transfor-

f.

mation

Let

S1

and

S2

be a pair of noncollinear vectors of

be the respective images of S L and vector of

and v

2

H,

S2

under the mapping

f.

If

H.

V

k t

is any

it follows from Theorem 4-10 that there exist real numbers

such that

V = vlSl

+ vZS2.

Since

f

vl

is a linear transformation, we

have

Accordingly,

Thus,

It follous that

f

is r e p r e ~ e n t e dby the matrix

when vectors are expressed in terms of their coordinates relatlve t o the basis

S1' S* You notice that the matrix A is completely determined by the e f f e c t of f on the pair of noncollinear v e c t o r s used as the basis for H. Thus, once you know that a given transformation on H is linear, you have a matrix representing the mapping when you have the imagee of the natural basis vectors,

For example, I t can be shown by a geometric argument that the counrerclock-

Isec* 5 3 1

198 0

wise rotation o f the plane through an angle of 30

rransformarion.

This function maps any point

measure of the angle POP' (Figure 5-11}

Figure >10.

P

about the origin is a linear onto the p o i n t

PI,

where the

(Figure 5-10). It is easy t o see

i s equal t o 30'

that

A rota tion through

en angle of 30'

Figure 5-1 1.

about the o r i g i n ,

(1,O)

The images of the points

and (0,l) under a r o t a t i o n of 30'

about the o r i g i n .

[;I

is rapped onto

[in:

$1

and

Thus, the matrix representing this rotation is

I.. COS

A =

m0

Note that the f i r s t column of A

the second c o l m n of

A

I.,

c..

[ ; 41 -

s i n 30'

30°

=

--

is the vector onto which

is the image o f

]

a

irmapped;

Ly1

The product or composition of two rransfowations is d e f i n e d just as you

[see.

5.31

define the composition of two real functions o f a real variable. Definition 5-3. vector

V

in

H

If

and

f

g

are transformations on

the composition transformations

Lg

and

then for each

H,

are the trans-

gf

formations such t h a t

fg(V)

f(g(V))

a

Thus, to find the image of

g,

apply f

maps

f.

and then apply

U onto

W,

then

V

gf(V) = g(f(V1).

and

under the transformation

Consequently, if

fg maps

V

maps

g

V

A, and

and

gf

onto

by

and i f

is a linear transformation represented by the matrix

f

are both linear transformations;

i s represented

U,

.

is a linear transformation represented by the matrix

g

you first

W.

onto

The following theorem is readily proved (Exercise 5-3--7)

Theorem S . If

fg,

is represented by

fg

B, AB,

then

fg

while

BA.

For example, suppose that in the coordinate plane each position vector is f i r s t r e f l e c t e d in the

Ln length.

y

axis, and then the resulting vector is doubled

L e t us f i n d a matrix representation o f the resulting linear tratls-

formation on

H.

If

g

i s the mapping t h a t transforms each v e c t o r into its

reflection in the vertical axis, then we have I Y I

I-

1-1

I

nl I Y l

If f maps each vector into twice the vector, then

Accordingly, the m a t r i x representing

fg

is

we have

gf

200 Exercises 5-3 1.

Show that each o f the mappings in Exerci a e 5-1-3

is linear, by determining

matrices representing the mappings. 2.

Conaider the linear transformations,

of

p:

reflection in the horizontal axis,

q:

hotizontal p r o j e c t i o n on the l i n e

r:

rotation counterclockwise through 90

a:

shear moving each p o i n t vertically through a distance equal to the abscissa of the p o i n t ,

y = 0

-x

(Exercise H - 5 e ) ,

,

H into H. In each of the following, determine the matrix represent-

ing the g i v e n transformation:

3.

Let

f

be the rotation of the plane counterclockwise through 45O about

the o r i g i n , and l e t P

g

be the rotation clockwise through . ' 0 3

matrix representing the rotation cmterc~oclruisethrough 15'

Determine

about the

origin. 4.

(a)

(b)

Prove that every linear transformation mape the origin onto i t s e l f . Prove that every linear transformation maps every subspace o f a subspace of

5.

f and g on H,

For every two linear transformations

be the transformation such that, for each (f

+ g)(V)

= f(V)

Mithou t using matrices, prove that

6.

For each linear transformation

af

H

onto

+g

to

H.

f

f

to be the transformation such that

H

f

V E H,

+ g(V1.

+g

on

define

is a linear trans formation on

and each real number

a,

define

Ii.

201 Without using matrices, prove that

is a linear transformation on

af

H.

7 . Prove Theorem 5-4. Without using matrices, prove each of the following:

8.

+ h)

(a>

f(g

(b)

(f

(c>

f(ae)

where

f,

+ g)h

= f g -4-

+ gh,

= fh

= a(fg),

and

g,

fh,

h

are any linear transformations on

and

H

a

i s

any real number.

5-4.

One-to-one

Linear Transformations

The reflection of the plane in the

axis clearly maps d i s t i n c t p o i n t s

x

onto distinct points; thus, the reflection is a one-to-one

linear transformation

on H. Moreover, the reflection maps any pair of noncollinear pair of noncollinear vectors.

It is easy t o show t h a t this property is common

H into itself.

linear transformations o f

to all one-to-one

Theorem 5-5, Every o n e t o a n e linear transformation on v e c t o r s onto

Proof.

onto a

vectors

H maps noncollinear

noncollinear v e c t o r s .

Let

S1

and

S2

f(S ) 1

be a pair of noncollinear vectors and let: =

T1

be their images under the one-to-one

one, we know that therefore, that

T1

TI

and

T, t

and

f ( S . 1 = T2

L

l i n e a r mapping

f.

Since

are not both the zero v e c t o r .

i s not the zero vector.

To show that

f

is one-to-

W e may suppose,

TI and T2

are

not c o l l i n e a r , we shall demonstrate that the assumption t h a t they are collinear

leads to a c o n t r a d i c t i o n .

If TI and that f

T2

= r T.I

T2 are collinear, then there exists Now, c o n s i d e r the image under

is linear, we have

f

a

real number

of the v e c t o r

r

r S1.

such

Since

202 ~ h u s ,each of the vectors

S2

is mapped onto

T2. Since

f

is one-

i t follows rhar

t-ne,

and therefore that the fact that

and

r S1 and

S1 and

S1 and

S2

are collinear v e c t o r s .

are not collinear.

S2

T2 are collinear must be false.

But this contradicts

Hence, the assumption t h a t

T~ f must m a p noncollinear

Consequently,

v e c t o r s onto noncollinear vectors.

Corollary 5-5-1. H

maps

is

Proof.

H

The subspace onto which a one-to-one

linear transformation

itself.

Since the subspace contains a pair of noncollinear vectors, the

corollary follows by use of Theorems 4-9 and 4-10. The link between one-to-one

transformations on

H and second-order

matrices having inverses i s given in the next theorem.

Theorem 5-6.

Then

A,

H

Let

f

be a linear transformation represented by the matrix

f

is one-to-one

if and o n l y if A

Proof.

Suppose that

A

having the same image under

has an inverse, f.

f(sl) =

Thus,

Hence,

and

has

an inverse.

Let

S1 and

Now,

AS1

and

f ( S 2 ) = AS2,

S2

be vectors in

Thus,

f

must be a o n e t o - o n e trans f o r m a t i o n .

On the other hand, suppose that

follows that every vector in l a r , there are vectors

W

H

and

f

i s

From Theorem 5-5, it

one-to-one.

i s the image of some vector in

U

H.

In p a r t i c u -

such that

and

Accordingly, the matrix having for its f i r s t column the vector

its aecond column the vector

U,

is the inverse

of

W,

and for

A.

A linear transformation represented by the matrix

Corollary 5-6-1.

A

is

o n e - - t ~ n e if and only i f

The theory of systems of tw linear equation8 in two variables can now be s t u d i e d geometrical t y

.

Writing the sys tern

V

t h a t are mapped by the matrix

i n the form

where

we seek the vectors

Isec.

3-4.3

A

onto the vector

U.

2@+

If 6(A) # 0 , onto

H.

we now know t h a t

Therefore,

A

A

represents a one-tc-one

- has

Thus, the s y s t e m (1) - or, equivalently, (2) If

F(A) = 0,

maps

A

a

and

Hence,

Xf

V =

A

U.

the coltrmns of

A

must

has the Ef rs t of these forms, then

A

In the o t h e r two cases,

H onto the o r i g i n .

namely,

m u s t have one of the forms

A

are z e r o .

b

U,

H -1

exactly one s o l u t i o n .

then, in v i r t u e of C o r o l l a r y 4-7-1,

be collinear v e c t o r s .

where n o t both

onto

V

maps exactly one vector

mapping of

[;I

line of vectors collinear with the vector

.

A

maps

(See Exercise

H o n t o the 54-7, b e l o w . )

With these results in mind, you may now complete the discussion of the s o l u t i o n

o f equation ( 2 )

.

Exercises 5-$

1.

Using Theorem 5-6 in Exercise 5-1-3

2.

or its corollary, determine which of the transformations are one-to-one.

Show thar a linear rransformation is o n e - t o o n e if and only if the kernel o f the mapping c o n s i s t s only of t h e zero vector.

3.

(a)

Show that i f

(See Exercise 5-2-12

is a o n e - t m n e linear transformation on

f

there e x i s t s a linear transformation

g

such that, for a l l

H,

.)

then

V c H,

and

The transformation

(b)

Prove that i f fg

i a called the inverse o f

Show that the transformation

transformation on 4.

g

-1

g = f

f

and is usually written

in part: (a) i s a one-tcrone

H.

f and

is also a one-to-one

g

are

onetrrone

linear transformations of

transformation of

H.

H,

then

205

5.

Prove that the s e t of one-t-ne

H

linear transformations on

i s a group

relative to the operation of composition of transformations. 6.

Show t h a t i f

are linear transformations o f H

f and g

i s a o n e t m n e transformation, then both

f

and

g

A

maps

such that

are onetc-one

fg trans-

forraations, 7.

(a)

Show that if

&(A) = 0,

then the matrix

H

onto a point

( t h e o r i g i n ) or onto a line. Shov that i f

A

i s the zero matrix and

then every vector

V

of

(b)

is

the zero vector,

is a s o l u t i o n of the equation AV = U.

H

Shov that if 6(A) = 0, but

(c)

U

A

is not the zero matrix, then the

A

is

solution s e t o f the equation

is a s e t of collinear vectors.

Show that if 6(A) = 0, but

(d)

not

the zero matrix,and

U

is

not the zero v e c t o r , then the solution set of the equation

either is empty or consists

where

V1

and

V2

of

are fixed vectors such that AV1 = U

8.

Show that if the equation

U,

55.

then

A

all vectors of the form

AV = U

and

AV2 =

[:I

has more than one s o l u t i o n for any given

does not have an inverse.

Characteristic Values and Characreristic Vectora If we think of a mapping as "carryingt' p o i n t s of the plane onto other points

o f the plane, we might ask, through curiosity, i f there are

cases

in which the

206 image p o i n t under a mapping

i s

the same as t h e p o i n t itaelf.

Such 'fixedf

p o i n t s , or vectors, are o f great importance in mathematical a n a l y s i s . Let us look at an example.

The r e f l e c t i o n with respect t o the

axis,

x

that i s ,

has the property of mapping each vector on the

axis onto i t s e l f ; thus,

x

each of these vectors is fixed under the transformation.

Definition 5 4 .

If a transformation of H

into itself maps a g i v e n vector

onto Ftaelf, then that: vector is a fixed vector for the transformation, More generally, we are interested in any vector that is mapped into a m u l t i p l e of i t s e l f ; that i s , we seek a vector

V

E

H and a number c E R

such

rha r

Since the equation is automatically s a t i s f i e d by the zero vector regardless of the value

c,

t h i s v e c t o r i s ruled o u t .

The number

the vector

V

c

is called a characteristic value (or eigenvalue)

a characteris t i c vector o f A.

A, and

of

These notions are fundamental in

atomic physics since the energy levels of atoms and molecules turn out to be

given by the eigenvalues of certain matrices.

A l s o the analysis o f f l u t t e r

and vibration phenomena, the stability analysis of an airplane, and many other physical problems require finding the characteristic values and vectors of

matrices.

In Section 5 1 , we

carried the plane

saw t h a t the mapping

H onto the line

y = x/2.

I f we consider the s e t

F

207 under t h i s same mapping, we see that

collinear with

F.

F

i s mapped onto

F'

, the

s e t of vectors

Note that

and hence t h a t 5 is a characteristic value associated with a characteristic vector for any

D e f i n i t i o n 5-5.

t c R,

t

#

A,

and

12:1 L

0.

4

Is

Each nonzero vector s a t i s f y i n g the equation

i s called a characteristic vector, corresponding to the characteristic value

(or characteristic root)

c

of

A.

Note that, as remarked above, the t r i v i a l s o l u t i o n ,

is not considered a c h a r a c t e r i s t i c vector,

[:I

of the equation

Because of the importance of characteristic values i n pure and a p p l i e d mathematics, we need a method for f i n d i n g them.

and real numbers

c

we seek nonzero vectors

such t h a t

If I is the i d e n t i t y matrix of order 2, then (1) can be written

If

we l e t

equation ( 2 ) becomes

as

V

v

We know t h a t there is a nonzero vector

s a t i s f y i n g an equation of the form

BV = 0 if and only if

Hence equation ( 2 ) has a s o l u t i o n o t h e r than the zero vector if and o n l y if is chosen in such

a

c

way as ro s a t i s f y the e q u a t i o n

Rearranged, this equation becomes

which is called the characteristic equation of the matrix quadratic equation is solved f o r

the corresponding vectors

c,

equation (1) can readily be found, as

Example.

Once t h i s

V

satisfying

illustrated in the following example.

Determine the fixed lines under the mapping

We must solve the matrix equation

Since

6(A

A.

- cI) = ( 2

[see*

- c)(l - c ) , 5-51

the characteristic equation is

the roots o f which are

1

and

2.

For

c = 1,

equation ( 3 ) becomes

which is equivalent t o the system

Thus,

A

maps the l i n e

is mapped onto i t s e l f . is invariant:

if V

x

+

3y = 0

onto i t s e l f ; t h a t i s , the s e t

Actually, since

i s a c b r a c t e r i s t i c vector,

then f(V)

For

c = 2,

Hence,

c = 1,

equation ( 3 ) becomes

-

V.

F

each vector of t h i s s u b s p a c e f(V)

i t s image, and

c = 1,

y = 0

maps the line

onto i t s e l f ; the s e t of vectors

F

is closed under the transformation.

The characteristtc equation associated with the matrix

This equation expresses a real-number function.

For a matrix function, the

corresponding equation i s

where

order 2.

If we substitute

we f i n d that

any

and 0 IB the zero matrix of in this matrix equation,

I is the i d e n t i t y matrix o f order

2 x 2

A

is

a

A

2

root of i t s characteristic equation, This La true f o r

matrix.

Theorem F7. The m t r i x

A

is a s o l u t i o n o f its characteristic equation

The proof is left as an exercise.

Theorem 5-7 is the case n = 2

o f a famoua theorem called the Cayley-

Hamilton Theorem, vhich s t a t e s that an analogous result holds for matrices of any order

n.

Exercises 5-5

1.

Determine the characteristic roots and vectors o f each of the follouing matrices:

2.

Prove that zero is a characteristic root of a matrix 6(A)

3.

= 0.

Show that a

linear transformation

f

is one-t-ne

if and only i f zero i s

not a characteristic root of the matrix representing 4.

if and only i f

A

f

.

Determine the invariant subspaces (fixed Lines) of the mapping given by

Show t h a t these l i n e s are m ~ c u a l l yp$rpendicular. is a solution of i t s characteristic 22

(matrix) equation

6.

Show that

but =hat 2

is

[i]

an

invariant vector of the transformation

is not invariant under this mapping.

23.2

7.

Show that

maps every line through che o r i g i n onto itself if and only if

A

A

8.

for

r # O .

Let

d = (al1 - a Z 2 )

L

real numbers.

+4

=

[: or]

a

where

12a21' Show rhat the number

a l l ' a12' =21' and aZ2 are any o f d i s t i n c t real characteristic roots

of the matrix

9.

Find a nonzero matrix t h a t leaves no Line through the o r i g i n f i x e d .

linear transformation that maps exactly one line

10. Determine a one-te-one

through the origin onto i t s e l f .

11,

Show rhat every matrix roots i f

12.

B

o f the form

# 0.

Show that the matrix

A

[::I

and i t s transpose

has two d i s t i n c t characteris tic

have the same characteristic

A~

roots.

5-6.

Rotations and Reflections

Since length is an important property in Euclidean geometry, we shall look

for the linear transformations of the plane that l e a v e unchanged the length I I VI I

of every vector

V.

Examples of such transformetions are the following:

(a)

the reflection of the plane in the

(b)

a r o t a t i o n of the p l a n e through any g i v e n angle about the o r i g i n ,

(c)

a reflection

x

axis,

in t h e x axis followed by a ra t a t i o n

about the o r i g i n .

I

2u A c t u a l l y , we can show that any linear transformation that preserves the lengths of a l l v e c t o r s i s e q u i v a l e n t t o one of these three.

The following theorem w i l l

be very u s e f u l i n proving t h a t r e s u l t .

Theorem 5-8.

A linear transformation of

H

that leaves unchanged the

l e n g t h of every vector also leaves unchanged (a) the inner product o f every pair

of vectors and ( b ) the magnitude o f the angle between every pair of v e c t o r s . Proof.

Let

V

and

U

be a pair of vectors in

be t h e i r respective imegea under the transformation. 4-Hi,

we

and l e t

H

Vt

and

U'

In virtue of Exercise

have

and

Since the transformation is linear, f o r the image of

Consequently, (2) can be written as

II(V

+u)'112

= IIV'II

2

+ 2Vi

+U

V

.

U'

+

we have

IIV'II

2

.

But the transformation preserves the Length o f each v e c t o r ; thus, we obtain

IIV'II = IIVII,

IIU'II = IIUII,

and

II(V f U)'II

-

IIV

+ Utl.

Making these s u b s t i t u t i o n s i n equation (3), ue g e t

Comparing equations (1) and ( 4 ) , you see that we must have

t h a t i s , t h e transformation preserves the value of the inner p r d u c t .

[see. 563

(3)

a4 Since the magnitude of the angle between

V

U

and

can be expressed in

terms of i n n e r products ITheorem 4-51, it f o l l o w s that the transformation also

preserves that magnitude.

If a linear trans formarion preserves the length of every

Corollary 5-8-1.

vector, 'then it maps orthogonal vectors onto orthogonal v e c t o r s .

By the d e f i n i t i o n o f orthogonality, t h i s simply means t h a t the geometric vectors are mutually perpendicular. X t is very easy to show the transformations we are considering a l s o pre-

serve t h e d i s t a n c e between every pair of points in the plane.

Ke state this

property formally i n the next theorem, the proof of which is left as an exercise.

A linear transformation that preserves the length of e v e r y

Theorem 5-9,

vector leaves unchanged the distance between every p a i r of p ~ i n t sin the plane;

V'

that i s , if

U'

and

are the respective images of the v e c t o r s

V

and

U,

then

Let us now f i n d a matrix representing any given l i n e a r lengttr-preserving

t r a n s f o m a t i u n of

All we need to f i n d are the images of the vectors

H.

under such e transformation.

If

Si

that both

and

Si and

(Why is this s o ? )

are the respective images of

S;

Si

are o f Length

L

SL

and that they

and

S2,

then we know

are orthogonal to each

other.

Suppose t h a t the

x

S;

axis (Figure 5-12).

We know that:

S;

(alpha) with the positive half o f

forms the angle

Since the length of

is perpendicular to

S;

.

S;

equals

Hence, there are

1,

we have

two o p p o s i t e

Figure 5-12

.

A length-preserving trans formation.

possibilities for the direction of

S;,

because the angle

makes with the positive.half of the

x

axis may be either

In the firs t case (5) , we have

I n the second case (61, we have n

[sec.

561

s i n Cr

P

(beta) that

Si

236 AccordingLy, any linear transformation

t h a t leaves the length of each

f

vector unchanged must be repreeented by a matrix having either the form cos

a -sin

[r

I

or the form

In

the f i r s t instance (71, the transformation

vectors

SL and

o f the entire plane

the v e c t o r

r = IIVII;

f

is a rotation

To v e r i f y t h i s observation, we write

through that angle.

H

in terms o f its angle a f inclination

V

and the Length

Forming

and we suspect that

through an angle

S2

simply rotates the basis

f

(theta) t o the

9

x

axis

that is, we write

from equations ( 7 ) and (9), we o b t a i n

AV

[

AV =

I

r(cos 9 cos - s i n 8 sin a) r(sfn 8 cos cX + cos 8 sin a)

'

From the formulas of trigonometry,

w e see

+ a) = ( 8 + a) =

cos ( 0

cos 8 cos

a - sin 8

sin

s i n 8 cos

(Y

+ cos

Q sin

a, a,

that =

[

is the vector o f length

r

A.

Thus,

AV

axis.

We have proved that the matrix

the angle

r cos (9 r s i n (8

+ a)

+ a)

a t an angle

A

8

+ Cr

to the horizontal

represents a rotation of

fl

through

(T.

But suppose

f

i s

represented b y the matrix

B

in equation (8) above.

This transformation differs from the one represented by S;

sin

is reflected across the line of the vector

S;.

A

in that the vector

Consequently, you may

suspect t h a t rhFs transformation amounts to a r e f l e c t i o n of the plane in the

[sec.

2-61

217 axis followed by a rotation through the angle

x

reflection in the

you may

x

therefore

Since you know that the

LY.

axis is represented by the matrix

expect that

We leave t h i s verification as an exercise.

Exercises 5 6

1.

Obtain the matrices that rotate

Write

through.the following angles:

(a)

180°,

(£1

(b)

45O,

(g)

(c)

30°,

(h)

(dl

boo,

i

(e)

2.

H

out

270°,

90°, -120°,

360°,

-13s0,

(j)

LSO'.

the matrices that represent the transformation conaieting of a

r e f l e c t i o n in the

x

axis

followed by the rotations of Exercise 1.

3.

Verify Equatian (lo), above.

4.

A linear transformation of H

t h a t preserves the Length of every vector is

called an orthogonal transformation, and the matrix representing the' transformation is called an orthogonal matrix.

Prove that the transpose of an

orthogonal m a t r i x is orthogonal.

5.

Show that the inverse o f an orthogonal matrix is an orthogonal matrix.

6.

Show that the product of two orthogonal matrices i s orthogonal.

7.

(a)

Show that a translation of

In the direction of the vector

and through a distance equal to the length o f

[sac. 5-61

U

is g i v e n by the mapping

(b)

Show that this mapping does not preserve the l e n g t h of every vector,

b u t t h a t it does preserve t h e d i s t a n c e between every pair o f p o i n t s in the

plane.

8.

(c)

Determine whether or n o t t h i s mapping is linear.

Let

Ra

and

respectively.

through

9.

B

denote rotations o f

(a)

H

through t h e anglea

Prove that a rotation through

0 amounts to a rotation through CI

Note t h a t the matrix

number. 0 .

R

A

Q

+ @;

a and

followed by a r o t a t i o n that

i s , show

that

of Equation (7) is a representation of a complex

What does the result o f Exercise 8 imply f o r complex numbers?

Find a matrix that represents a reflection across the line of the

vector

(b)

b,

Show that t h e matrix

B

of equation

flection across the Line of some v e c t o r .

(a),

above, represents a re-

Appendix RESEARCH EXERC X S E S

The e x e r c i s e s i n t h i s Appendix are essentially "research-type" problems d e s i g n e d to exhibit aspects of theory and practice in macrix algebra t h a t c o u l d n o t be included in the t e x t .

They are especially suited as i n d i v i d u a l assign-

ments for those students who are p r o s p e c t i v e majors i n the theoretical and

p r a c t i c a l aspects o f the scientific d i s c i p l i n e s , and for students who would like

AlternativeLy, small groups of s t u d e n t s

to test their mathematical powers. might join forces in working them.

1.

Quaternions.

was invented by the

The algebraic system that is explored in this e x e r c i s e

Irish mathematician and physicist, William Rowan Hamilton,

who p u b l i s h e d his first paper on the subject in 1835.

It was nor until 1858

that Arthur Cayley, an English mathematician and lawyer, p u b l i s h e d the f i r s t

research paper on matrices, though the nane matrix had p r e v i o u s l y been a p p l i e d by James Joseph S y l v e s t e r in 1850 t o rectangular arrays of numbers.

Hamilton's

s y s t e m o f quaternions i s

Since

a c t u a l l y an a l g e b r a o f matrices, i t is more

easily presented i n this guise than in the form in which i t was first developed.

In the present exercise, we shall consider the algebra o f w i t h complex numbers as e n t r i e s .

and inversion remain the same.

and we denote by

where

z, w, zl,

K

and

K

matrices

The definitions of a d d i t i o n , m u l t i p l i c a t i o n ,

We use C

for the set of all complex numbers

the s e t o f a11 matrices

w

L

are elemerlts o f

having real entries, the element

of

2 x 2

has an inverse if and only if

C. As i s the case w i t h matrices

220 and then we have

Since

1

is a complex number, the unit matrix is s t i l l

If z

a

x 4- iy,

then we write

and call this dumber the complex conjugate of

A quaternion i~ m element

q

[G q] , We denote by (a)

Q

6 ( q ) = x2

Hence conclude that, since

q =

or simply the conjugate of

of the particular form

r t C

and

w E C.

the s e t o f all quaternions.

Ohow that

and only i f

of K

z,

+ y 2 + uZ + v2

x, y , u,

and

v

if

r

-

x

+ iy

are real numbers,

and

w = u

6(q) = 0

if

2.

Show that if q E Q, then q has an inveree if and only if q and exhibit the form of q-' if it exists. (b)

Four elements o f names :

+ iv.

Q

+ 0- ,

are of particular importance and we g i v e them special

z.

(c)

where

Show t h a t if

z

5

x f iy

and

w ='u

+ iv,

then

q =X + yIU + u v * v w .

(d)

Prove the following i d e n t i t i e s involving

I, U,

V

and

W:

v'PV2iJi-f and

UV=W=-W,

and

VW=U=-WV,

(el Use the preceding two exercises t o show then q + r , q - r, and q r are all elements of

W=V--uw.

that if

q

E Q

Q.

The conjugate of the element

q =

and the

norm and

[

1,

where r = x

+ iy,

v

-

u

+ lv,

trace are given respectively by

and

(f)

Show that i f

q f Q,

and i f

q

is invertible, then

and

r

E Q,

222 9, and if q-l

From t h i s c ~ n lcu d e that if q -Z

(g)

Show that each q

(h)

Show that i f

6

q

-1

r 9.

s a t i s f i e s the quadratic equation

Q

q E Q,

cnis t a , then

then

Note t h a t t h i s may be proved by using the result t h a t i f

then

and then ueing the results given in ( d ) . ( i ) Show t h a t i f q E Q

and

r e Q,

then

tqrl = Iql I r l

and

The geometry o f quaterniona constitutes a very i n t e r e s t i n g subject. requires the representation of a quaternion

as a p o i n t w i t h coordinates

(a,

b, e, d)

in four4imenaional spaces.

It

The

subset of elements, QL

{q: 9

E

Q

and

Iql = I),

is a group and is represented geometrically as the hyperephere w i t h equation

Nonassociative Algebras

2.

The algebra o f matrices (we r e s t r i c t our attention in t h i s exercise to the set

M

of

cation.

2 x 2

matrices) has an associative but not a comutative multipli-

"Algebras" with nonassociative multiplication have become increasingly example, in mathema tical gene t i c s .

important in recent years-for

Genetics is

a eubdiacipline o f biology and is concerned w i t h transmission o f hereditary

traits.

Nonassociative "algebras" are important also in the study of quantum

mechanics, a subdiscipline o f p h y s i c s .

We give first:

a simple

algebra (named a f t e r the Norwegian geometer Sophus LLe) If

A E M

and read this (a)

and

"A op B,"

"op"

(i)

AoB =

- BOA,

(ii)

AoA =

Q,

(iv)

being an abbreviation f o r operation. o:

Ao(BoC)

+ Bo(CoA) + Co(AoB)

AoI =

= IoA.

G i v e an example ro show that

and hence that

o

i s not

example of a L i e

we write

Prove the following properties of

liii)

(b)

8 E bf,

.

=

Ao(BoC)

2,

and

(AoB)oC

are d i f f e r e n t

an associative operation,

D e s p i t e these strange properties,

o

behaves nicely relative to ordinary

metrix addition. (c)

Show that

o

dietributes over addition:

(d)

Show that

o

behaves nicely r e l a t i v e to multiplication by a number.

and

224 It

will be recalled that

and is defined as the

element

A'

is termed the multiplicative inverse of

B

A

satisfying the r e l a t i o n s h i p s AB =

-

I

BA.

But i t m u s t elso be recalled thar t h i s d e f i n i t i o n was motivated by the fact that A1 = A =

that is, by the fact t h a t (e)

IA,

I is a multiplicative unit.

Show rhar there is no

o

unit.

thar

We know, from the foregoing work,

o

i s

neither c o m u t a t i v e n o r

Here is another kind of operation, called Jordan multiplication:

associative.

If A f M and

wedefine

B f M,

AjB =

(AB

+ BA) 2

-

We see at once that AjB = B j A ,

so that Jordan mulriplicarion is a comutative operation; b u t i t i s n s

associative

(f)

.

Determine all of the properties o f the operation

For example, does

3.

j

j

that you can.

d i s t r i b u t e over a d d i t i o n ?

The Algebra of Subsets

We have seen that there are interestfng algebraically defined subsets of

2 x 2 matrices.

One such subset, f o r example, is the set

M,

the set of all

Z,

which is isomorphic wf th the set of complex numbers.

Much of higher

mathematics i e concerned w i t h the "global structure" of "algebras," and generally this involves the consideration of eubsets of the "algebras" being s t u d i e d . t h i s exercise, we shall generally underscore l e t t e r s t o denote subsets of If

4

and

g are

subsets of

M,

then

In

M.

is the s e t of a l l elements of the form

where

A + B ,

A E

4 and B E -B.

In set-builder notation this may be written

A + B = -

{A+B

* By an a d d i t i v e subset of

: A s 4

and

B E B].

is meant a subset

M

A

CM

such that

Determine which of the f o l l o w i n g are additive subsets o f

(a)

(i)

M:

(213

(ii> (11,

(iii)

M,

(iv)

2,

(v)

%,

(vi)

the s e t of a l l

(i) if B=

(ii)

(Lii)

A+

0,

-

and if

with

x

(d)

E

2

with

are subsets of

A -

1,

M,

then

of

M,

then A - + C-C B-+ C -.

C -B

& and B are addttive subsets

M.

denote the s e t o f a l l column vectors

R

and

Show t h a t i f

5

M whose entries are nonnegative.

V

y c

&(A)

( A + B ) +C,

=

is also an addittve subset of

Let

and

M

!+A,

(B + C-)

Prove t h a t if

(c)

in

t h e set of all elements of

Prove t h a t if A,

(b)

A

R.

v

is

a

fixed element of

V,

then

then

A: A E M

M.

is an a d d i t i v e subset of

set

=

Notice also that i f

If A - and B are subsets

is the

Av

and

M,

of

AV = 0

then

(-A)"

=

0.

then

of all AB, A

and

E

B

E

g.

Using set-builder notation, we can w r i t e this in the form AB =

A subset

A

of

M

IAB: A

6 A -

and

B

E

01.

is multiplicative if

(e)

Which of the s u b s e t s in p a r t (a) are multiplicative?

(f)

Show t h a t i f

(ii)

and i f

A -,

B, -

ACE,

and

are subsets of

C -

then

M,

then

A C C BC. -

A

B of -,.

M

such t h a t

(g)

Give an example of two s u b s e t s

(h)

Determine which o f the following subsets are multiplicative:

and

(iii)

the s e t of a l l elements o f

M

w i t h negative entries,

(iv)

the s e t o f all elements of

M

f o r which t h e upper left-hand

M

o f t h e form

e n t r y is l e s s than 1, (v)

the set of all elements of

with

Osx,

Osy,

x + y