Linear Dependency and Independency

 

11   Line Linear ar dependence dependence and and independen independence ce Definition: A Definition:  A finite set S  set  S    = { x1 , x2 , . . . , xm }  of vectors in   Rn is said to be  linearly dependent   if there exist scalars (real numbers)   c1 , c2 , . . . , cm , not all of which are 0, such that

c1 x1 + c2 x2 + . . . + cmxm  =  0 . Examples:

1. The vectors

       1   1   3  x  =  1  ,   x  =  −1  ,   and   x  =  1        1

2

3

1

2

4

are linearly dependent because 2x1 + x2 − x3  =  0 . 2. Any set containing the vector   0  is linearly dependent, because for any  any   c  = 0, c0  =  0 . 3. In the definition, we require that not all of the scalars c scalars  c 1 , . . . , cn  are 0. The reason for this is that otherwise, any set of vectors would be linearly dependent. 4.  If a set of vectors is linearly dependent, then one of them can be written as a linear combination of the others:  (We just do this for 3 vectors, but it is true for any number). = 0 (that is, supppose the the  c i   Suppose c Suppose  c 1 x1 + c2 x2 + c3 x3  =  0 , where at least one of the c vectors are linearly dependent). If, say,  say,   c2   = 0, then we can solve for   x2: x2  = (−1/c2)(c )(c1 x1 + c3 x3 ).

And similarly if some other coefficient is not zero. 5. In principle, it is an easy matter to determine whether a finite set   S   S   is linearly dependen pend ent: t: We wri write te down down a sys system tem of lin linear ear algeb algebrai raicc equ equati ations ons and see if the there re are solutions. For instance, suppose

   1 S   =   2  ,   1

   1 1 ,   =  {x , x , x }.       1  0 1

−1

1

1

2

3

 

By the definition, S  definition,  S  is  is linearly dependent   ⇐⇒  we can find scalars c scalars  c 1 , c2 ,   and and   c3 , not all 0, such that c1 x1 + c2 x2 + c3x3  =  0 . We write this equation out in matrix form:

  1  2

1 0

 1   c 1  c 

1   −1 1

1

2

c3

     0   =  0  0

Evidently, the set  set   S  is   is linearly dependent if and only if there is a non-trivial solution to this homogeneous equation. Row reduction of the matrix leads quickly to

 1  0

1 1

 1   .

  1

2

0 0 1

This matrix is non-singular, so the only solution to the homogeneous equation is the trivial one with c with  c 1  = c  =  c2  = c  =  c3  = 0. So the vectors are   not  linearly dependent.

Definition Definition:: the set S  set  S    is   linearly independent   if it’s not linearly depen dependen dent. t. What could be clearer? The set S  set  S  is  is not linearly dependent if, whenever some linear combination of the elements of   S  S  adds  adds up to   0, it turns out that   c1 , c2 , . . .  .   are   all    zero. zero. Tha Thatt is, example le abov above, e, we assume assumed d c1 x1  + · · · + cn xn   =   0   ⇒   c1   =   c2   =   · · ·   =   cn  = 0. In the last examp that c that  c 1 x1 + c2 x2 + c3 x3  =  0  and were led to the conclusion that all the coefficients must be 0. So this set is linearly independent. The “test” for linear independenc independencee is the same as that for linear depende dependence. nce. We set up a homogeneous system of equations, and find out whether (dependent) or not (independent) it has non-trivial solutions

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Exercises:

1. A set S  set  S  consisting  consisting of two different vectors   u  and   v  is linearly dependent   ⇐⇒   one of  the two two is a non nonze zero ro mu multi ltiple ple of the othe other. r. (Do (Don’t n’t forge forgett the possib possibili ility ty that one of  the vectors could be   0). If neit neither her v vec ector tor is   0, the vectors are linearly dependent if  they are parallel. parallel. What is the geome geometric tric condi condition tion for three nonzero ve vectors ctors in   R3 to be linearly dependent? 2. Find two linearly independent vectors belonging to the null space of the matrix

  A  =  

3

2   −1

1

0

−2   −2

2

 4  3 . 

3   −1

3. Are the columns of   A  (above) linearly independent in   R3 ? Wh Why? y? Ar Aree the rrow owss of   A linearly independent in   R4 ? Why?

Elementary tary row row operation operationss 11.1   Elemen We can show that elementary row operations performed on a matrix  A don’t  A  don’t change the row space. We just give the proof for one of the operations; the other two are left as exercises. Suppose that, in the matrix A matrix A,, rowi (A) is replaced by row i (A)+ c · row j (A). Call the re resultin sultingg matrix B matrix  B.. If   x  belongs to the row space of  A,  A , then x  = c  =  c1 row1 (A) + . . . + ci rowi (A) + . . . + c j row j (A) + cm rowm (A).

Now add and subtract c subtract  c · ci  · row j (A) to get

x   =   c1 row1(A) + . . . + ci rowi (A) + c · ci row j (A) + . . . + (c (c j  − ci  · c)row j (A) + cm rowm (A)

=   c1 row1(B ) + . . . + ci rowi (B ) + . . . + (c (c j  − ci  · c)row j (B ) + . . . + cmrowm (B ). 3

 

This shows that   x  can also be written as a linear combination of the rows of   of   B . So aan ny element in the row space of  A  A  is contained in the row space of  B.  B . of   B   is contained in the Exercise:   Show the converse - that any element in the row space of   row space of  A.  A . Definition: Definition: Two  Two sets X  sets  X    and and Y   Y    are  equal   if   X   ⊆  Y   Y    and and Y   Y   ⊆  X   X .. This is what we’ve just shown for the two row spaces. Exercises:

1. Show that the other two elementary row operations don’t change the row space of  A of  A.. 2. **Show that whe when n we m multiply ultiply any matrix matrix A  A by  by another matrix B matrix B on  on the left, the rows of the product B product  BA A  are linear combinations of the rows of   A. 3. **Sho **Show w that when we m multip ultiply ly A  A on  on the right by B by  B,, that the columns of  AB are  AB  are linear combinations of the columns of  A  A

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