Matrix 2
September 15, 2022 | Author: Anonymous | Category: N/A
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Description
Fatrcx Fj. Altar Lafao Asscstaht Wrmnkssmr mn Fahabkfkht N@U,@[W
Fatrcx Ajjcmh / Uu`traemh /
Fuopoceamh9 @mml- @mwkh (Wabk-173)
(13). Bcvkh tdk nmoomwchb fatrceks A6
: 0 1 4
Nchj
(a) >A ― ;@
@6
> 11
2
0
(`) -5E + 0J
E6
; 1 3 > 7 11
(e) AE + @J (j) EA + @J
10
J6
17
2 0:
1;
4
Fatrcx (07).
Bcvkh tdk nmoomwchb nmoomwchb fatrceks fatrceks
> 2 11 A6 1 10 0 17 : 4
Nchj
@6
07 1; ; > 4 17 : 5 1
E 6
7 > 0 : 2 : > 7 ;
(a) 5A - 0@ (`) -:E + ;J (e) AE + @J
J6
(j) EA + J@
; 4 2 > 5 0 1; 10 :
Zdk Jktkrfchaht mn a Fatrcx • Zdk jktkrfchaht mn a fatrcx A cs emffmhoy jkhmtkj `y |A| mr jkt A. • Jktkrfchahts kxcst mhoy nmr squark fatrceks. • Zdky ark a fatrcx edaraetkrcse (tdat eah `k smfkwdat tkjcmus tm emfputk).
Zdk Jktkrfchaht nmr a 0x0 Fatrcx • Cn wk davk a fatrcx A sued tdat A
tdkh A
a11 a10 a01 a00
6 a11a00 - a10a01
Nmr kxafpok, tdk jktkrfchaht mn A
cs
A
1 0 5 :
6 1 0 6 a11a00 - a10 a01 1 : - 0 5 6 -0 5 :
Jktkrfchahts nmr 0x0 fatrceks ark kasy!
Zdk Jktkrfchaht nmr a 5x5 Fatrcx • Cn wk davk a fatrcx A sued tdat A
a11 a10 a0511 a0500
a15
a0555
Zdkh tdk jktkrfchaht cs jkt A 6
A
6 a11 a00 a50
a05 - a10 a01 a55 a 51 55
a05 + a15 a01 a51 a55 55
a00 a50
wdced eah `k kxpahjkj ahj rkwrckh as jkt
A
6
A
55 50 05 51 6 a11a00a 55 - a11a05a 50 + a10 a05 a51 - a10 a01 a55 + a15a01 a50 - a15a00 a51
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Cn wk rkwrctk tdk jktkrfchahts nmr kaed mn tdk 0x0 su`fatrceks ch jkt A 6
as
A
A
00 6 a11 a a50
05 05 01 a a 10 a a55 a 51 55
05 05 01 a + a 15 a a51 a55 55
a00 a50
a05 6a00 a55 - a 05 a50 , a55
a01 a51
a 05 6a01a55 - a05a51 , ahj a 55
a01 a51
a 00 6a01a50 - a00 a 51 a 50
`y su`stumh wk davk
00 00 a a50
6 a11a00 a55 - a11a05 a50 + a10a05 a51 - a10a01a55 + a15a01a50 - a15a 00a51
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Hmtk tdat cn wk davk a fatrcx A sued tdat A
a11 a10 a01 a00
a15 a05
a51 a50
a55
Zdkh |A| eah aosm `k wrckh as jkt A 6
mr jkt A 6
A
A
6 a11
a00 a50
a05 a 01 05 - a 10 a55 a 51 55
6 -a01
a10 a50
a15 a + a00 11 a 55 a 51
a10 a00
a15 a11 15 a05 - a50 a01
a05 a01 05 + a 15 a a55 51 55
a00 00 a50
a05 a11 - a 05 a55 a51
a10 a50
a15 a11 15 a05 + a55 a01
a10 10 a00
mr jkt
A
6
A
6 a51
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Zm jm sm rst erkatk a fatrcx mn tdk safk jcfkhscmhs as A emhscshb mhoy mn aotkrhahb scbhs (+,-,+,…) + - + - + - + -
+
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Zdkh kxpahj mh ahy rmw rmw mr emoufh (c.k., fuopoy kaed kokfkht ch tdk skoketkj rmw/emoufh `y tdk emrrkspmhjchb scbh, tdkh fuopoy kaed mn tdksk rksuots `y tdk jktkrfchaht mn tdk su`fatrcx tdat rksuots nrmf kocfchamh mn tdk rmw ahj emoufh tm wdced tdk kokfkht `komhbs Nmr kxafpok, okt–s kxpahj mh tdk skemhj emoufh
a11 A
a10
a15
01 00 a a a51 a50
05 a a55
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Zdk tdrkk kokfkhts mh wdced mur kxpahscmh cs `askj wcoo `k a10, a00, ahj a50. Zdk emrrkspmhjchb scbhs ark -, +, -. + - + - + - + - +
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Um nmr tdk rst tkrf mn mur kxpahscmh wk wcoo fuopoy -a10 `y tdk jktkrfchaht mn tdk fatrcx nmrfkj wdkh rmw 1 ahj emoufh 0 ark kocfchatkj nrmf A (eaookj tdk fchmr ahj mkh jkhmtkj A wdkrk r ahj e ark tdk jkoktkj rmws ahj emoufhs)9 re
A
11 10 a a a01 a00 00 a51 a50
wdced bcvks us
15 a a05 sm a55
-a 10
A10
a01
a05
a51
a55
a01 a05 a51 a55
Zdcs prmjuet cs eaookj a emnaetmr .
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Nmr tdk skemhj tkrf mn mur kxpahscmh wk wcoo fuopoy a00 `y tdk jktkrfchaht mn tdk fatrcx nmrfkj wdkh rmw 0 ahj emoufh 0 ark kocfchatkj nrmf A9 A
a11 a10 a01 a00 00 a51 a50
a15 a05 sm
a55
A 00
wdced bcvks us a00
a11 a51
a15 a5 5
a11 a51
a15 a55
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Nchaooy, nmr tdk tdcrj tkrf mn mur kxpahscmh wk wcoo fuopoy -a50 `y tdk jktkrfchaht mn tdk fatrcx nmrfkj wdkh rmw 5 ahj emoufh 0 ark kocfchatkj nrmf A9
A
a11 a10 01 00 00 a a 51 50
a15
a0555 sm
A
wdced bcvks us -a50
a11 a01
a15 a05
00
a11 a01
a15 a05
Zdk Jktkrfchaht nmr a 5x5 Fatrcx Wuhb tdcs aoo tmbktdk tmbktdkrr yckojs jkt A 6
A
6 -a 10
a01 a 51
a05 a + a00 11 a a 55
a15 a - a50 11 a a
51
55
01
a15 a 05
Um tdkrk ark hchk jcshet ways tm eaoeuoatk tdk jktkrfchaht mn a 5x5 fatrcx! Zdksk eah `k kxprksskj as f
jkt A 6
A
6 aci -1 i61 i6 1
c+ i
h
A ci
6 aci -1
c+ i
A ci
c61 c6 1
Hmtk tdat tdcs cs rknkrrkj tm as tdk fktdmj mn emnaetmrs emn aetmrs ahj eah `k uskj tm hj tdk
jktkrfchaht mn ahy squark squark fatrcx.
Zdk Jktkrfchaht nmr a 5x5 Fatrcx ― Ah Kxafpok Uuppmsk wk dav davk k tdk nmoomwchb fatrcx A9 1 A 6 0
0 ;
5 :
1 -5 -0
[schb rmw 1 (c.k., c61), tdk jktkrfchaht cs9 f
jkt A 6
A
6 a1i -1
1+i
A 1i
1(0) 0( 4) 5( 11) 1;
i61 i6 1
Hmtk tdat tdcs cs tdk safk rksuot wk wmuoj aedckvk uschb ahy mtdkr rmw mr emoufh!
Fatrcx • EM-NAEZMPU MN A FAZPCT 5 Wrm`okf9 Cn A6 0 2
: ; 5
> 3 2
Nchj tdk em-naetmrs em-naetmrs mn kokfkhts mn > ahj - 3.
5 Umoumh 9 Zdk em-naetmr mn > c.k.A05 6(-1)0+5 2
5 Ahj tdk emnaetmr mn -3 c.k. A55 6 (-1) 0
•
5+5
: 5 6
-1( 3 ― 04 )613
:
6 1; +4 6 05 . ;
,
AJIMCHZ MN A UQ[APK FAZPCT
Wrm`okf9 Nchj tdk ajimcht mn tdk fatrcx ahj vkrcny tdk tdkmrkf A6 A(AjiA) 6 (AjiA)A 6 A C 5
1 1 1 1 0 5 0 1 5
Fatrcx Zdk em-naetmr mn A ark A11 6(-1)
1+1
A15 6(-1)
1+5
A00 6(-1)0+0
1 0 0 1
1 0
1 A516(-1)5+1
0
A55 6(-1)
5+5
1 5 6 - (5+>) 6 - 3 0 5
0 5 6 > -5 6 5 A10 6(-1)1+0 1 5 6 (-1- :) 6 - ; A01 6(-1)
0+1
1
6 5
(5- 0) 6 1
1
6 (-5-0) 6 - ; 5
1 1 6 1 0
(0-1) 6 1
A05 6(-1)0+5
1 1
1
6 - (5+1) 6 - : 5
1 1 6 - (-1-0) 6 5 0 1
1 A50 6(-1)5+0
1
1 5
6 - (-5-1) 6 :
Fatrcx t 5 : ; 5 3 ; 3 1 : Aji A 6 : 1 5 6 ; 5 1 ; : 1 1
1
O.D.U. A(AjiA) 6 1
0
0 1
5 3 ; : 1 5 ; : 1 6 5 14 1; : 0 3 ; 4 5 1 > 3 1; 4 1 3 17 : 5
1 5 : ; : 5 3 1 5 ;
5
1 7 11 7 6 7 11 7 6 -11 7 7 7 7 11
7
7
1
7
7
1
5 : ; 1 1 1 : 3 1 1 0 5 6 F.D.U. (AjiA) A 6 1 0 1 5 ; 5
6
1 7 11 7 7 7 11 7 6 -11 7 7 7 11
6 -11 C5
54; 5 10 1; 5 : 17 3 1 17 3 0 : 3 5 10 ; 5 0 ; > 1 ; 3 5
7
7
1
7
7
1
6
-11 C5
Fatrcx P.D.U. 6 A C5 6
1
1
1
1
0
5
0
1
5
0
C5 6\1 1
5 5
1
-1
0
5 5
+1
6\1(>-5) -1(5 + >) +1(-1 -:)V A
6 \5 - 3 ― ;V C5 6 -11 C5 Dkhek, O.D.U 6 F.D.U 6 P.D.U (^krckj) Wrm`okf9 Emfputk tdk chvkrsk mn tdk fatrceks A 6
Rk lhmw, A-1 6 AjiA / |A|
--- ----- (1)
1 1 1 1 0 5 0 1 5
C5
1
0
0
1
V C5
ZDK CH^KPUK MN A FAZPCT
Ch mrjchary arctdfke fuopoceamh tdk chvkrsk mn 5 cs ahj tdk chvkrsk mn 1 cs 5 schek 5
5
1 5
1 ahj
1 5
1
5 1
5
Zdk chvkrsk mn a fatrcx A cs emffmhoy jkhmtkj `y A-1 mr chv A.
Zdk chvkrsk mn ah h x h fatrcx A cs tdk fatrcx A-1 sued tdat AA-1 6 C 6 A-1A
Fatrcx
Wrm`okf9 Emfputk tdk chvkrsk mn tdk fatrceks A
Rk lhmw, A-1 6 AjiA / |A|
--- ----- (1)
1 1 1 6 1 0 5 0 1 5
Fatrcx Zdk em-naetmr mn A ark A11 6(-1)
1+1
A15 6(-1)
1+5
A00 6(-1)0+0
1 0 0 1
1 0
1 A516(-1)5+1
0
A55 6(-1)
5+5
1 5 6 - (5+>) 6 - 3 0 5
0 5 6 > -5 6 5 A10 6(-1)1+0 1 5 6 (-1- :) 6 - ; A01 6(-1)
0+1
1
6 5
(5- 0) 6 1
1
6 (-5-0) 6 - ; 5
1 1 6 1 0
(0-1) 6 1
A05 6(-1)0+5
1 1
1
6 - (5+1) 6 - : 5
1 1 6 - (-1-0) 6 5 0 1
1 A50 6(-1)5+0
1
1 5
6 - (-5-1) 6 :
Fatrcx 5 : ; 5 3 ; t 3 1 : Aji A 6 : 1 5 6 1 ; : ; 5 1
Hmw,
A 6
1
1
1
1 0
1
0
5 5
0
6 1
1
5 5
-1
0
5 5
+1
1
0
0
1
61(>-5) -1(5 + >) +1( -1 ― : )
65 ― 3 ―; Nrmf kqh.(1) wk davk, A-1 6
5 : ; : 3 1 ; 5 1
-11
6 -11 (Ahswkr)
Hmtk9 Rk mhoy emhscjkr chvkrsks mn 0Ò0 fatrceks dkrk
Kxafpok 1
Nchj tdk chvkrsk mn
ù1 ù-1
Ahj sdmw ct cs a ok chvkrsk
0ù :ù
Kxafpok 1
Uwap ù1
ùù1 0ù ù ù 1 :ù
ù 1
1
: ù 0 ù1
0ù 1ù
ù1 : 1 jktkrfchaht
Edahbk scbh
Kxafpok 1
ù1
ùù1 0ù ù ù 1 :ù
ù 1 >
ù: ù1
0ù 1ù
Hmtk9 Aotdmubd, strcetoy spkalchb, tdk chvkr chvkrsk sk sdmuoj davk tkrf arcschb nrmf tdk jktkrfchaht fuopockj ch, ch praeek wk mkh okavk tdcs tkrf mut ch nrmht mn tdk fatrcx.
ù Kxafpok 1
Udmw ct cs a ok chvkrsk
-1
ù: ù 0ùù 1 0ù ù ùùù ù 1ùù 1 :ù ù1 1 ù > 7ù ù ù ù > ù 7 >ù ù 1 7ù ù ù 7 1ù
ù 1 0ù ù 1 0ù 1 ù ù ù ù ù ùù1 : ù ùù1 :ù >
Fatrcx • Ucfuotahkmus Kquamh • Nmrf 9- AT 6 @ T 6 A-1@ Rdkrk A cs tdk fatrcx mn em-keckht T cs tdk emoufh vketmr mn uhlhmwhs ahj @ cs tdk emoufh vketmr mn emhstahts. emhstahts.
Kxafpok- (@mml lapmmr9pabk-4:2)9 [schb fatrceks smovk emfpoktkoy tdk nmoomwchb kquamhs ;x ― >y + :z 6 1; 2x + :y ― 5z 6 13 0x + y + >z 6 :>
Fatrcx Umoumh9 Zdk systkf eah `k wrckh ch fatrcx nmrf as
; > : x 2 : 5 y 6 > 0 1 z Cn wk emfpark emfpark ct tm Ax6
1; 13 :> ` nmrf
; > : A 6 2 : 5 T 6 > 0 1
1; x y @ 6 13 :> z -1@ Rk davk, Ax6 @ x 6A x 6 (AjiA)@ / |A| --- ---(1) ; > : 2 : 2 5 : 5 +: |A| 6 2 : 5 6 ; + > 0 1 1
0
1
>
0
>
>
6 ;(0:+5 ) +>(:0 + >) +:(2-4)
6:13
Fatrcx Zdk em-naetmr mn A ark A11 6(-1)1+1
2 0
A15 6(-1)
1+5
A00 6(-1)0+0 A516(-1)
A55 6(-1)
:
1
; : 6 0 >
6 (2- 4) 6 - 1 A01 6(-1)
0+1
(57- 4) 6 00
> : 6 (14-1>) 6 0 : 5
5+1
; > 6 2 :
5+5
2 5 6 - (:0+>) 6 - :4 0 >
: 5 6 0:+5 6 02 A10 6(-1)1+0 1 >
(07+:0) 6 >0
A05 6(-1)0+5
> : 6 - (-5>-:) 6 :7 1 >
; 1
; A50 6(-1)5+0
2
>
6 - (;+10) 6 -12 1 : 6 5
- (-1;-04) 6 :5
Fatrcx AjiA 6
:7 0 02 :4 1 t 02 :7 00 12 6 :4 00 :5 0 :5 >0 1 12 >0
Nrmf kquatcmh (1), T 6
x y 6 z
:7 0 02 :4 00 :5 1 12 >0
:13
1 :13
@
:7 02 :4 00 1 12
10;2 5 1 6 :13 1>2> 6 : > 0;1:
Dkhek x65 , y 6: ahj z 6 >
1; :5 13 >0 :> 0
Chvkrsk `y Ucfuotahkmus Kquamhs • Wrk mr pmstfuopoy ymur squark fatrcx A `y a juffy fatrcx mn tdk safk jcfkhscmhs, c.k., AA
1
a11 a10 a01 a00 a51 a50
a15 a ` e a05 j k n a55 b d c
Ukt rksuot kquao tmas ahymur cjkhty fatrcx mn tdk tdk safk jcfkhscmhs squark fatrcx A, c.k., AA 1
a11 a10 a01 a00 a a
a15 a ` e 1 7 7 a05 j k n 7 1 7 a b d c 7 7 1
51
50
55
Chvkrsk `y Ucfuotahkmus Kquamhs • Pkembhczk tdat tdk rksuohb kxprksscmh cfpocks a skt mn h0 scfuotahkmus kquamhs tdat fust `k saskj cn A-1 kxcsts9 a11(a) + a10(j) + a15(b) 6 1,
a11(`) + a10(k) + a15(d) 6 7,
a11(e) + a10(n) + a15(c) 6 7< mr a01(a) + a00(j) + a05(b) 6 7, mr
a01(`) + a00(k) + a05(d) 6 1,
a01(e) + a00(n) + a05(c) 6 7< a51(a) + a50(j) + a55(b) 6 7,
a51(`) + a50(k) + a55(d) 6 7,
a51(e) + a50(n) + a55(c) 6 1.
Umovchb tdcs skt mn h0 kquamhs -1
scfuotahkmusoy yckojs A .
Chvkrsk `y Ucfuotahkmus Kquamhs ― Ah Kxafpok • Cn wk davk
1 0 5 A 6 0 ; : 1 -5 -0
Zdkh tdk pmstfuopockj fatrcx wmuoj `k AA
1
1 0
0 ;
5 a ` e : j k n
0 b d 1 -5 --0
c
Rk hmw skt tdcs kquao tm a 5x5 cjkhty fatrcx 1 0
0 ;
5 j a ` 7 7 : k en 1 7 1 7
1 -5 -0 b d
c
7 7 1
Chvkrsk `y Ucfuotahkmus Kquamhs Ah Kxafpok Pkembhczk Pkembhczk tdat tdk―rksuohb kxprksscmh cfpocks tdk nmoomwchb h0 scfuotahkmus kquamhs9 1a + 0j + 5b 6 1, 1` + 0k + 5d 6 7, 1e + 0n + 5c 6 7< mr 0a + ;j + :b 6 7, 0` + ;k + :d 6 1, 0e + ;n + :c 6 7< mr 1a - 5j - 0b 6 7, 1` - 5k - 0d 6 7, 1e - 5n - 0c 6 1.
Zdcs systkf eah `k saskj c A-1 kxcsts.
Chvkrsk `y Ucfuotahkmus Kquamhs Ah Kxafpok Umovchb tdk skt mn h―0 kquamhs scfuotahkmusoy yckojs9 a 6 -0/1;, ` 6 1/5, e 6 2/5, j 6 -4/1;, k 6 1/5, n 6 -0/5– b 6 11/1;, d 6-1/5, c 6 -1/1; sm wk davk tdat A-1 -1
A
-0 1;
1
5
6 -41; 1 5 11 -1
2
1;
-01; -1
1;
5
1;
Chvkrsk `y Ucfuotahkmus Kquamhs Ah Kxafpok AORASU edkel ymur―ahswkr. Dmw= [sk tdk naet tdat AA-16 A-1A 6C ahj jm a ocok fatrcx fuopoceamh! 1 2 -0 5 1; 1 7 7 1 0 5 1; -1 -0 7 1 7 C AA 6 0 ; : -4 1 5x 5 1 -5 -0 111; -15 -11; 7 7 1 1; 5 1;
Um wk davk nmuhj A-1!
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf • Aubfkht ymur fatrcx A wctd ah cjkhty fatrcx mn tdk safk jcfkhscmhs, c.k., a11 A | C a01 a 51
a10 a00 a50
a15 1 7 7 a05 7 1 7 a55 7 7 1
Hmw wk usk vaocj Pmw Mpkramhs hkekssary tm emhvkrt emhvkrt A tm C (ahj sm A|C tm C| A-1)
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf • ^aocj Pmw Mpkramhs mh A|C • Smu fay chtkredahbk rmws • Smu fay fuopoy a rmw `y a seaoar • Smu fay rkpoaek a rmw wctd tdk suf mn tdat rmw ahj ahmtdkr rmw fuopockj `y a seaoar (wdced cs mkh hkbavk)
• Kvkry mpkramh pkrnmrfkj mh A fust `k pkrnmrfkj mh C • [sk vaocj Pmw Mpkramhs mh A|C tm emhvkrt A tm C (ahj sm A|C tm C|A-1)
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok • Cn wk davk
1 0 5 A 6 0 ; : 1 -5 -0
Zdkh tdk aubfkhtkj fatrcx A|C cs 1 0 5 1 7 7 A | C 0 ; : 7 1 7 1 -5 -0 7 7 1
Rk hmw wcsd tm usk vaocj rmw mpkramhs tm emhvkrt tdk A scjk mn tdcs aubfkhtkj fatrcx tm C
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Utkp 19 Uu`traet 0²Pmw 1 nrmf Pmw 0 0 ; : 7 1 7 - 0 1 0 5 1 7 7 7 1 -0 -0 -1 7
Ahj su`stutk tdk rksuot nmr Pmw 0 ch A|C 1 0 5 1 7 7 7 -15 -0 -70 1 7 7 1 1
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Utkp 09 Uu`traet Pmw 5 nrmf Pmw 1
1 - 1 7
0
51
7
-5
-0 7
7
;
;1
7
7
1 -1
Jcvcjk tdk rksuot `y ; ahj su`stutk nmr Pmw 5 ch tdk fatrcx jkrcvkj ch tdk prkvcmus stkp
1 0 5 1 7 1 -0 -0
7 1
7 7
7 1
1 1
;
7
1 ;
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Utkp 59 Uu`traet Pmw 0 nrmf Pmw 5
7 7
7
1
1 1
1
-0 - 0
1
7
5 11
-1
;
;
7
; 7
-1
-1
;
Jcvcjk tdk rksuot `y 5 ahj su`stutk nmr Pmw 5 ch tdk fatrcx jkrcvkj ch tdk prkvcmus stkp
1 0 5 1 7 1 -0 -0
7 1
7 7
7 7
1 11 1;
1
5
1 1;
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Utkp :9 Uu`traet 0²Pmw 0 nrmf Pmw 1 1 0 5 1 0 7 1 -0 - 0
1 7
7 7 1 7
2 ; -0 7
Uu`stutk tdk rksuot nmr Pmw 1 ch tdk fatrcx jkrcvkj ch tdk prkvcmus stkp 1 7 2 ; 7 1 -0 -0
-0 1
7 7
7 7
1 11 1;
1
5
1 1;
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Utkp ;9 Uu`traet 2²Pmw 5 nrmf Pmw 1 1
7
2
;
-0
2 7
7
1 11 1;
-1
1
7
7 -0
1
5
1;
7
-1 1; 2
5
1;
Uu`stutk tdk rksuot nmr Pmw 1 ch tdk fatrcx jkrcvkj ch tdk prkvcmus stkp
-0 1 7 7 1; 7 1 -0 -0
1
5 1
2
1; 7
7 7
1 11
1;
1
5
1
1;
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Utkp >9 Ajj 0²Pmw 5 tm Pmw 0 7
1
0 7
7
1 11 1;
-1
7
1
7 -4
1
-0 - 0
1;
7
1
5
-1 1; -0
5
1;
Uu`stutk tdk rksuot nmr Pmw 0 ch tdk fatrcx jkrcvkj ch tdk prkvcmus stkp -0 1 5 1 7 7 1; 7 1 7 -4 1; 1 5
1; -01; 2
7 7 1 11 -1 -1 1; 5 1;
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Hmw tdat tdk ok scjk mn tdk aubfkhtkj fatrcx cs ah cjkhty fatrcx C, tdk rcbdt scjk mn aubfkhtkj fatrcx cs tdk chvkrsk mn A (A-1), c.k., tdktdk fatrcx -0 A
1
1
2
-4 1; 1 5 -01; 1; 5 1; 11 -1 -1 1; 5 1;
Chvkrsk `y tdk Bauss-Imrjah Aobmrctdf ― Ah Kxafpok Zm edkel mur wmrl, okt–s skk cn mur rksuot yckojs AA-1 6 C9
AA
1
1 2 -0 5 1; 1 0 5 1; -0 0 ; : -4 1 1 -5 -0 111; -15 -11; 5 1; 1;
Um mur wmrl edkels mut!
1 7 7 7 1 7 7 7 1
Chvkrsk `y Jktkrfchahts • Pkpoaek kaed kokfkht aci ch a fatrcx A wctd ah kokfkht eaoeuoatkj as nmoomws9 • • • •
Nchj tdk jktkrfchaht mn tdk su`fatrcx tdat rksuots wdkh tdk ctd rmw ahj itd emoufh ark kocfchatkj nrmf A (c.k., Aci) Aaed tdk scbh tdat ymu cjkhkj ch tdk Fktdmj mn Emnaetmrs Jcvcjk `y tdk jktkrfchaht mn A Akr aoo kokfkhts davk `kkh rkpoaekj, trahspmsk tdk rksuohb fatrcx
Chvkrsk `y Jktkrfchahts ― Ah Kxafpok Abach suppmsk wk davk smfk fatrcx A9 A
1 0 5 0 ; : 1 -5 -0
Rk davk eaoeuoatkj tdk jktkrfchaht mn A tm `k ―1;, sm wk rkpoaek kokfkht 1,1 wctd A 11 1 11
1 0 1; 01; A
Ucfcoaroy, wk rkpoaek kokfkht 1,0 wctd 110 A10
1 4
4
A
1;
1;
Chvkrsk `y Jktkrfchahts ― Ah Kxafpok Akr uschb tdcs apprmaed tm rkpoaek kaed mn tdk hchk kokfkhts mn A, Zdk kvkhtuao rksuot wcoo `k
-01; -4 1;
15 1 5
2 1; -0 1;
111;
-1 5
-11;
wdced cs A-1!
AWWOCEAZCMH MN FAZPCT Wrm`okf9 Fr.. T cs Fr c s a smok trajkr trajkr,, fahunaeturchb fah unaeturchb ta`oks ahj edacrs. Kaed ta`ok rkqucrks ; dmurs mn oa`mr ahj > uhcts mn fatkrcao. A edacr rkqucrks 5 oa`mr dmurs ahj 5 uhcts mn fatkrcao. Cn Fr. T poahs tm prmjuek 17 ta`oks ahj 1; edacrs ch tdk hkxt wkkl, dmw fahy oa`mr dmurs wcoo dk hkkj tm wmrl ahj dmw fued fatkrcao wcoo dk rkqucrk=
AWWOCEAZCMH MN FAZPCT Zdk fatrcx smoumh wmuoj `k9
Za`oks Edacrs
17
1; 1 0
Oa`mur
; 5
Fatkrcaos >
5 0 0
Oa`mur 6
F Fa atkrcaos
3;
17; 1 0
Ahs9 3; dmurs wcoo `k hkkj tm wmrl wmrl ahj 17; uhcts fatkrcao fatkrcao wcoo `k rkqucrkj rkqucrkj
AWWOCEAZCMH MN FAZPCT • A pu`ocsdchb dmusk das twm `rahedks. Ch kaed `rahed, tdkrk ark tdrkk meks. Ch kaed mek, tdkrk ark 5 pkmhs, : eokrls ahj ; typcsts. Ch mhk mek mn a `rahed, > saoksfkh ark aosm wmrlchb. Ch kaed mek mn mtdkr `rahed 0 dkaj-eokrls ark aosm wmrlchb. [schb fatrcx hmtamh hj c) tdk tmtao huf`kr mn pmsts mn kaed lchj ch aoo tdk meks talkh tmbktdkr ch kaed `rahed, cc) tdk tmtao huf`kr mn pmsts mn kaed lchj ch aoo tdk meks
talkh tal kh tmbktdkr nmrf `mtd tdk `rahedks.
AWWOCEAZCMH MN FAZPCT • A hahek emfpahy das meks omeatkj ch ; jcvcscmhs 57 jcstrcets ahj 077 tahas. Kaed mek das 1 dkaj eokrl, 1 easdckr,, 1 eokrl, ahj 1 pkmh. A jcvcscmhao mek das, ch easdckr ajjcmh, ah mek supkrchtkhjkht, supkrchtkhjkht, 0 eokrls, 1 ttypcst ypcst ahj 1 pkmh. A jcstrcet mek das, ch ajjcmh, 1 eokrl ahj 1 pkmh. Zdk `asce fmhtdoy saoarcks ark as nmoomws9 mek supkrchtkhjkht Zl 17,777, dkaj eokrl Zl 4777, 4 777, easdckr Zl2777, eokrl, ahj typcst Zl >777 ahj pkmh Zl;777. [schb Fatrcx hmtamh hj9 c) tdk tmtao huf`kr mn pmsts mn kaed lchj ch aoo tdk meks talkh tmbktdkr, cc) tdk tmtao `asce fmhtdoy saoary `coo mn kaed lchj mn mek, ahj ccc) tdk tmtao `asce fmhtdoy saoary `coo mn aoo tdk meks talkh tmbktdkr.
AWWOCEAZCMH MN FAZPCT Wrm`okf9 Zdk tmtao emst mn fahunaeturchb tdrkk typks mn fmtmr ear cs bcvkh `y tdk nmoomwchb ta`ok9
Oa`mur ((d drs)
Fatkrcaos (uhcts)
Uu`-emhtraetkj (uhcts)
Ear A
:7
177
;7
Ea Ear @
47
1;7
47
Ea E ar E
177
0;7
177
Oa`mur emsts Ps. 07 pkr dmur, dmur, uhcts mn fatkrcao emst Ps.; kaed ahj uhcts mn su`-emhtraetkj wmrl emst Ps.17 pkr uhct. Nchj tdk tmtao emst mn fahunaeturchb 5777, 0777 ahj 1777 vkdceoks mn typk A, @ ahj E rkspkevkoy.
AWWOCEAZCMH MN FAZPCT Umoumh9 Okt fatrcx W rkprkskhts oa`mur dmurs, fatkrcao uskj ahj su`-emhtraetkj wmrl nmr tdrkk typks mn ears A, @, E rkspkevkoy :7 W6 47 177
177 1;7 0;7
;7
47 177
Okt fatrcx Q rkprkskht rkprkskht oa`mur emst pkr uhct, fatkrcao emst emst ahj emst emst mn su`-emhtraetkj wmrl 07 Q6
;
17
AWWOCEAZCMH MN FAZPCT Zdk emst mn kaed ear A , @, E cs hmw bcvkh `y tdk emoufh fatrcx
WQ 6
:7 47 177
177
;7
1;7
177
0;7
47
07 ; 17
1477 6 51;7 :0;7
Okt tdk huf`kr mn ears A , @, E tm `k fahunaeturkj ch tdat mrjkr `k rkprkskhtkj `y tdk rmw fatrcx P6
5777
0777
1777
AWWOCEAZCMH MN FAZPCT Dkhek tdk tmtao tmtao emst mn fahunaeturchb tdrkk ears A @ ahj E cs bcvk `y tdk fatrcx
(WQ)P 6
6
1477 51;7 :0;7
5777
1,;3,;7,777
0777
1777
AWWOCEAZCMH MN FAZPCT Wrm`okf9 A,@ ahj E das Ps. :47 , Ks 2>7 ahj Ps. 217 rkspkevkoy . Zdky uoczkj tdk afmuhts tm puredask tdrkk typks mn sdarks mn prceks x,y ahj z rkspkev rkspkevkoy koy.. A p puredasks uredasks 0 sdarks mn prcek x, ; sdarks mn prcek y ahj a hj 5 sdarks mn prcek z. @ puredasks : sdarks mn prcek x, 5 sdarks mn prcek y ahj > sdarks mn prcek z. E puredasks 1 sdarks mn prcek x, : sdarks mn prcek y ahj 17 sdarks mn prcek z. [schb fatrcx, hj tdk prceks pkr uhct mn tdk tdrkk emffmjcks.
AWWOCEAZCMH MN FAZPCT Umoumh9 Rk m`tach tdk nmoomwchb skt mn sufuotahkmus ochkar kquamhs 0x +;y + 5z 6 :47 :x +5y + >z 6 237 T + :y+ 17z 6 217
AWWOCEAZCMH MN FAZPCT • A fahunaeturchb emfpahy prmjueks tdrkk typks mn tkokvcscmh skts A, @, E. Zdk nmoomwchb fatrcx sdmws sdmws tdk saok mn tkokvcscmh skts ch twm jckrkht ecks9 :77 577
577
077
077
177
Cn emst prcek mn kaed skt A, @, E cs Zl 1777, Zl 0777, Zl 5777 rkspkevkoy ahj skoochb prcek Zl 1;77, Zl 5777, Zl :777 rkspkevkoy, Nchj tdk tmtao prmt uschb fatrcx aobk`ra
AWWOCEAZCMH MN FAZPCT A spkecao nmmj nmr atdoktks cs tm `k jkvkompkj nrmf twm nmmjs9 nmmjs W ahj nmmj Q. Zdk hkw nmmj cs tm `k jkscbhkj sm tdat ct emhtachs kxaetoy 1> muheks mn vctafch A, kxaetoy :: vctafch @ ahj kxaetoy 10 muheks mn vctafch E. Kaed pmuhj mn nmmj W emhtachs 1 muhek mn vctafch A, ; muheks mn vctafch @ ahj 1 muhek mn vctafch E. Mh tdk mtdkr dahj, Kaed pmuhj mn nmmj, Q emhtachs 0 muheks mn vctafch A, 1 muhek mn vctafch @ ahj 1 muhek mn vctafch E. Nchj tdk huf`kr mn pmuhjs
mn kaed nmmj tm `k uskj ch tdk fcxturk ch mrjkr tm fkkt tdk a`mvk rkqucrkfkht.
Kxafpok Umovk tdk nmoomwchb kquamh `y uschb fatrcx fktdmj9 x+17y+:7z6>3;7 x+3y+5;z6>20; x+10y+:7z62177
Wrm`okf Ah afmuht mn $>;,777 cs chvkstkj ch tdrkk chvkstfkhts at ratks mn >%, 4% ahj 3% pkr ahhuf rkspkevkoy. Zdk tmtao ahhuao chemfk cs $:,477. Zdk chemfk nrmf tdk tdcrj chvkstfkht cs $>77 fmrk tdah tdk chemfk nrmf tdk skemhj chvkstfkht. chvkstfkht. [schb fatrcx aobk`ra, jktkrfchk tdk afmuht mn kaed chvkstfkht
Wrm`okf • A emfpahy prmjuek tdrkk prmjuets kvkry jay. Zdkcr tmtao prmjuemh mh a ekrtach jay cs :; tmhs. Ct cs nmuhj tdat tdk tmtao prmjuemh mn tdcrj prmjuet kxekkj tdk tmtao prmjuemh mn rst prmjuet `y 4 tmhs wdcok tdk tmtao prmjuemh mn rst ahj tdcrj prmjuet cs twcek tdk prmjuemh mn skemhj prmjuet. Jktkrfchk prmjuemh okvko mn kaed prmjuet `y uschb
fatrcx aobk`ra fktdmjs.
Okt, tdk prmjuemh okvko mn tdk tdrkk prmjuets `k x, y ahj z rkspkevkoy. Zdkh, x+y+z6:; Y6x+4 mr, mr, -x+7y+z64 T+z60y T+z 60y mr, mr, x-0y+z67 x-0y+z67
Wrm`okf Ah autmfm`cok emfpahy usks tdrkk typks mn stkko s1, s0 ahj s5 nmr prmjuechb prmjuechb tdrkk typks mn ears e1, e0 ahj e5. Utkko rkqucrkfkht rkqucrkfkht (ch tmhs) nmr kaed typk mn ear ark bcvkh `komw9 Utkko
Ears
e1
e0
e5
s1
0
5
:
s0
1
1
0
5
5
0
1
Jktkrfchk tdk huf`kr mn ears mn kaed typk wdced eah `k prmjuek uschb 03, 15 ahj 1> tmhs mn
Utkkos mn tdrkk typks rkspkevkoy.
Zdk prceks ch tala tala mn tdrkk emffmjcks T, S ahj Y ark x, y ahj z pkr uhct rkspkevkoy. rkspkevkoy. A puredask : uhcts mn Y ahj skoos 5 uhcts mn T ahj ; uhcts mn S. @ puredask 5uhcts mn S ahj skoos 0 uhcts mn T ahj 1 uhcts mn Y. E puredask 1 uhct mn T ahj skoos : uhcts mn S ahj > uhcts mn Y. Ch tdk prmekss A, @ ahj E karh Zl >777, Zl;777 ahj Zl 15777 rkspkevkoy. [schb fatrcx fatrcx hj tdk prcek pkr uhct mn tdk tdrkk
emffmjcks.
5x+;y-:z6>777 0x-5y+z6;777 -x+:y+>z61577
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