3.2-3.4 - Mafi; 43/ ~2453/‘3nmenf 4/ So/uhbns...

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Unformatted text preview: Mafi; 43/ ~2453/‘3nmenf 4/ So/uhbns S‘ecfi‘on 2-5 M (—1.3), (2,4)} m fi'= {(0qu, (SM a —1) +5 2 = "Q'fzb 3 'I 34—!» (0‘): ~a+2b =7, ~q+21>=o=701=2b 34—19210 =9 saw—bun --=> Sb=lo -? _._...6=2~ a=2(2)=’-! (5 __ ~q+2b __ _q+2b=5 o '(Ba—b "'9 3a-b=0 => b=3a =? ~a+2(30~)= 5 =7 54:5 ‘27 .4. =’ 3 b=3([)=3 50 (0,10% H(—I,3>+ 2(2,~1) (5,0): I(—I,3) + 3(2,~I) 756 change of zoom/mafia ma+n'x +11”. Chang es/ggcoordmmtes in“ ~ c/‘nauées :3 fl coor ; ~ H I Z. 3 In 1R2, let I. be +he line 5: mac , where m #0. final 4,, “Mayan Far T099), where 71's #19. ref/ec-l-fon. of fl 2 about L , 5ian (km) [5 on H18 [1'06 y=mx; T(/,m)= (Cm). Noflce 9: urfix {3 perpendicular to yzmx’ and (ma, I) :19 on graix m . 50 T(‘mJ’): -(‘mll): (mj’l) . Lei/8's? be qn ordered 6615/5be IR" and. {fl}: = Lei} be fhe 5+anc/qrd ordered 645:3 for ,I/f and m; 42 be ,Lhe ma'fn'x Mat chanjesfl'—coardlha£-e: fOfi-caarc/inaéesj ,Ic, Q: I “m __ _. I (m ,).738rz_ (Q ’—M2+, Since {Tl}: = 62" CT]; 61 , Hm ET]j = 62 [T] , <52“ => [T133= (0:23! (Maw; : (AM) (2:2 ance )3 .‘s +he flan dare! 0rd€red 545/3, #101 'T is laps mu/H/a/z'ca'llbn by [T]; _ 7Eereforg for any (1,5)6 IR; z MLH (2:1 : l (I~m1)x+2mg > Mz-H ZMX + (Ml-[)3 =7 T093): “42+! ((l-m2)x+ 2M3, ZmX-I- ("42-05) @ Claim $04 arm/B are Sinai/4r nxn mam‘ces, Men #04).— fir(3)_ Pmofl [a ,4 4/1513 be Similar nxn Mair/"€65. 75617 7% ere exist-5 4n. inverfib/e ma+rix 62 such firms B= (9",4CQ. 7564 #03) = if“ (CW/V32) = if (CGY'AMZ) = er (a2 ((9%)) 5 by Exem‘se l3 offec.2.3‘ :tr'( (MM/D :éf‘ (In A) = t (A . tr(13)= firm). r ) @ PFOVE H16 COM/lazy Weorem 2023 QED—[Lag AG Mnxn and an ordered basis fir mam [LAja/=&”‘A&, where 6Q is fhe. nxn ma4rl.x WAD-re‘jfflh CD/umn his the J'th VEC-for of M Assume five sat up. Let/8 bEMe seam/arc! ordered basis for F“ (: iQ.,€z)...,€n3). 758/1 £19 Wear-em 2,1551) [LA-:1); :A 6) Lef (Qbe ,lhe (Aarye of coordinate main‘x #vaf: chanjes ficoordinaEES {ofl—Loordinaées. Wen CQ is indeed fhe 17er maérix whose jf“ column is floeJ'th vedor 0,5 2/. And/by 7220mm. 2.23 (my: <9" {Lu} (:2 :&"A (Q I by Q) above. a Jed/on 3. I Q A: 1 Z 3 I B: I O 3 C: l O 3 ‘ O ‘ t ’ t —2 t ’ 0 —2_ -.7_ \S' -: I I i -3 I z} _3 ’ z" Addzbg —Z Hines Co/umn I +0 column 2 +mnsfofm5 Ayn-£5 3 Adding ’! Hm€5 WW, +0 row 2 fransfarms g [min C. A seiuence a; elemem‘ag/ ape/ah‘ans flmé Hans/arms C in-lv if ,‘s x o 3 “Rem? t o- 3 433-442 x o 3 (0 ~2. v2.) ’ o —2. ~2 7 o t o l “3 E 0 —3 —2. o -3 -2. 2. ([4131: If a ma+rix 62 can bc oéfainez/ ,4-om a ma+n‘x P 1%, 4r, 8/8”, 301%,} ‘ row Opera—Hon, +hen Pam be. 017+al‘n84 from C? by an Heme/27447 maJrix af 7663 54M€ bypE. 5335 .~ Lag/0 be a mm‘rix And lei (52 be a named): warned from P by an elemem‘ar] row OPE/tiff”) of type, k (ragga) 717817 5} Weorem 3.!) 3 an e/emenfag matrix 5 5‘6. 62:50. By 7Aeorem 3.2., E [5 :‘nven‘x'A/e and g” ,‘5 of Hoe Same éype as E , 118- gape/fl. man CQ=£P =7 E"(Q = E“E F => E"CQ=P Five/ore P can be obfm‘n 64’ from (Q by 5') 4r, e/emenfqy maM‘X of 19/6 k. 56¢ 25.01? 3 ., ,2. -lfis+R2 I I 0 fiz+R$ ‘ 0 o .1“: ‘ O O 0 0 ’ 0 0 I o 0 I 7Z8 Corollary +0 7560er 3-9 5+a+es {hat elem en+ary row and co/umn alum-Iran: Fraserve rank, and 71780er 3.5 sfaées 7%qu 1‘58. rank of Ant}, matrix eiuals the maxinaumnumber of 1465 final-{7 ind EPQNCIEnE column; , rank (,4): 3 l 2. O \ I > I Z 0 I ‘ 2 LI I 3 0 drum; 0 O I 3 ‘2 W A: 3 6 Z- 5 '3FH"33 0 0 Z 2. «2. _Rz+83 "‘1'3 ‘ ‘3 ’ 43.1%, o o 1 1 5 *1 t 2. o l ! ' z O I ’ > o O O O 2' t o 0 0 O ‘ 33* ‘I O O O 0 31 $3” ‘ 0 o o 0 g ‘ 2’ O ’ ‘ 3 I 2 O I o O O I ; ~2. vfig+fi, ' o O l I 0 o o o o ‘ 233W; 0 o o 0 I O 0 O O o o o 0 o 0 We aee Hmé Columns / 3 5 are linear/(7 independent, 50 rank (A): ( 2. Z ‘ _..._.._.; z z 7 .. .. "i 2— "Rfi'fig ,l w, .Z’RVHKZ O A“ 2. 5 o 0 o 0 rank (,4):3, But c/fm (Fa): ‘/ So ran/<64) ~‘7‘ JI’M(FL’) :7 ,4 ,‘5 nm‘ I'm/edible. @ T= P2(R)*?F2(F)) defined by TCFOO): (2cm F’bc) Leffi: x, 96-] be {'58 art/Wed basis fir Ira). ‘ ~TO) = (mu) (0) = 0 7170-? Owl) (I) = x+l 773(2) = (90”) (27C) :: 2 9(1er 3) :(g s g :27 rank 2 75 3 :- dI-M =7 no-l [nyfrfi'b /6 . o o 2 :7 T ,3 ML inven’ib/e (final-’5). T: IRS—“7P; CIR) defined by 7'(m,a2_,613)= (q,+a;+ (133+ (q'— 42 +43)x + 4' x1. Laug: 3(1’ 0/0) / (02/10); (0/ 0/1)} 52 a basis 76f £3 7.01010): /'I-)c+;~ca TCO,I,0) = 1-); TCO/O/I) = /+x so [7]}: (Ii-I () Now NEW/Y/com/ufe CUB-I I 00 Ill lo 0 _________7 ! I f (0 0 __..____9 w: ; o I 0 49mm; 0‘20 ’5 ‘ 0 $32. I o O O x ~R,+g3 0 ~I~I -! 0 l ‘y l ‘0 0 > I l 1 ‘00 7 0‘ 0 ‘/-‘/0 3+3 0! 0 yl’yzo ~1R; (0—44 402‘) z 3 Oo-x +71! \ (3 ‘ x0 9 I to 5’2.”/2.‘ 7 (at O VZ‘YLO -p‘g‘rg. o 1 O y; “y; 0 ‘RJA‘R‘ 001%,)24 00‘ 72. 2*! ...
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3.2-3.4 - Mafi; 43/ ~2453/‘3nmenf 4/ So/uhbns...

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