Qucsin 1. (1 pids Circle he crrec anwers (1 pir e.lch. Nlrrk -,nly ifhc sae enrl}ilr.s, be nle wihu furh assumpins. JusilicaiD is n reeded. bu increc lswels carrl- a 1 pinl, Dcralv. s ralrdrn grressing des r help. Yu n1ay leavc ary ans:wer blank (0 pins. yu *.ill als r gc a negrive al s{rre n alrv grdp f fivc quesids. f he ccfllcien marix. has a pir. psiix ir everv rw. l,hcr he sysem Ax : b has a uniqre slulix. e-< C'JLu,\,. 'f,e r'i\l hc rarge f he lincar ralsfrmain x +,x is he cluidl space f A. / f hc clumrs f an n x r xral span R". heu hev a-re linerrlv nrllependenr.. -u i.., i' * :n... f,r"rrr',/ rllarri,.\.8.-.qpha\p\abcr arbrar ('( -' " c-(q' : ll::rr irfrurs i,, R" i lin^ffly "p.na.nu. \;n ir,urrcius m,,rp rh, rr \p.rr: f he linear syscm,x : b is incmiser. he.n he ccllicicll [radx,.les l have a piv psiin id every clullrrr. e\ /.w he linear svsem,x : 0 has mre ha.n ne shrin wlcrever here are frcc r-ari:rbies. f S is a lincnrlv rlcpended cllecin f vecrs. hed each vccl,r ir,9 is a linear cmbinair f he her vecrs,- a r- L,: he nullspace f a x marix mus ccmain infinielr. manv vecrs. * f,6 e f,isa,1 x,1 xr*rix ard he sysl,cm,x:e, is r,rnsi:ren lr nah veur,,f ihe s ird.,r,l L,jbi-er...crlRr. rl.u isinvrihl^. i,_ he di,i,.,isr,,.l h- r,,w :l,di ^ ju,j l,, ulu,n..n"; l;,,',,., rrr,'r.,'*,^, re rrv in lhc cchclr fnn f hc auguered marix f a sl,sc[r is [0 0 0 1 0] hed hc sysed is incnsisen. f a flnie scb S' f vecrs spans R". hen sme srbse f S frms a ba^sis f Rr. lhe rnap?: R + R v,'hicjr reflccs pils abu he linc A:1 is ]inear. he firs rw f a mari{ prddc AB is he ils rw f, muliplied n Lc rish by,
Quesin. (1 pins. 10+ (a r hc mauix, bclw, fnd bases f he rullspae. clum space_ rw space dnd lef nullspdre. Make sure yur mehd js clcar. (b r wha vau* f, b des [,,,6] lie in rhe rw space? E].?ain. - z q lc l ar 00i RrqR,' b,rr-." (J(A'pr,"1 cj* 'ra' "6 "' P"- le = gr' r.,r"s i^ r'q- (A -- 11.a.: lz s l+ z p O l l0 q'l c l-e R.-R, r-r" -l -6r 1-8 ( R,'R1 R,(r f ll OO 6 H i..( 00 l -l / - = - \ -6!'1. "? LNul (A= rei c'iasp{ -l q b.': " r ul ( ls exg,e,.?l n +,,ns df,<.,"ri x 8 -n x, - l 8rr= C lc 8, - --, - {j - aaq l 6 yr+ y,! (x{-l -6x, x 0 0a Y- X 1z - l D r.,\ll - f,]-\' r''rr l c - 8 b a l0 0r O R1+8P, l 0l 0l 06 \^ A ' 'b+11 RrR. q-;e -1. e i, Q\ (A, \s "1 *l Ln c*ss]er -6 { S - r-6..^l '- r. P".(n 1".: q=9 =6 " "-!= 6 *j [-l<
Quesin. (10 pirs A linear ra.r0sfrmain? : R -+ R sends [1,]? [1,,]" and [,]," b [,, ]r. (a ind (b ind "([,6]". a vecr v R'? wih : [,8,]?. r else explair why n flrch vecr exiss "(v, a R. r-] = L-i ; q ^ f r" L".. i(l= r(, f-nri-[ r(r= A 6-l!rrR. rl - rl lul A f1 lr-l= l l-i L - r- r -l - [r a:r -l rl l l- l-l r [:l rl - zl: 1 -, L j * (i=,i'[i]-l"l ( := r(l-il;' f il ;lj'. n? A rl rg 6 ( --.1 6 l 1- l 0l L ll l -. R,-R. --., ^ -l _r -.,] L r("-=[ 1? --l!l /\'v - 1-0 { N R*P, l- r lr L,,.a\ v e "1". r', r lhc.$ s'enl E l,l + R.-P" l 0 8 i] r* -'"- e ki: Le c "use 1le q"+'" a=[ lcci-> hrcl, "
Quesin. (1 pus Deermine he invcrs f he fllwing marix by ne f w mehds: ciher by rw-reducin' r by using deerminars. Check yur answe-r. [ur,,1 ^f, A,l l-1 z :, 6-1 R--qe, lq sq rll L llc l-l R:-*,' a.: --8 -- ^:l'^ lz rl '-, r' l ^p R' *J Q R",ar-' Rr l LLi 6 c, 8 r9 -i :Q' - 6-1 '-6 1 1. al -{ ^? -- R- gb- a?r R;SR: l, { 0 -l -- R:. <- as ^^l O 0-l U U' -? Rr- Rn Rr- R. J - -q l -1-8 A.L l u -\ S 8 l - - Ll -.l {- -11 1 q q -?'8+6 l-.\ji lr, s' -,{+ l - 6 - \6', -8 -+\- -1 9 18-\0 fl r!-16 + O