exam4sol - 1 In the vector space M2X2]R let S be the...

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Unformatted text preview: 1. In the vector space M2X2(]R), let S be the subspace consisting of symmetric matrices, and let U be the subspace consisting of matrices A such that (1 1)A(})=<o>. Find a basis of 3 F] U. in \ Ii A15 sawhfiwj ”‘2 ECLW J“ qwi‘m DQALWHO) W5 a+1b+c=o, Thug/re AeSnU FR (6.5.0 :9 M J94. l-oQimsn‘r/KJ Space 4 SoLub‘dMS t4 a+Zb+c=o> wi‘fh CW1 -\wv Mamba \wcflapemow‘i “Loni-ions Pwvtooiunj a baCt‘S, 2.3. i<<1)-0«3, ii) i. 2. Show that if there exists an invertible matrix A E Mnxn(R) such that AT = *A, then n must be even (AT denotes the transpose of A). CW9 "M 5:8“ ‘4 023”” CW5 ’ike 9‘sz «fl JoiM), w CW”) m 93“ 4 an «4 m5, we 3% de£(*/-\)= commune. Nm dai-(AT)=&L+(A), So 4 AT=-A, MM 0M(A)=(~u)‘“o&+m. 1; A x; aim-Ram exam #0, So we Ccu. Gama I93 cam) +9 @214 (-‘Y‘Zi, .22., M is Wm. 3. Let T: V —) V be a linear transformation. Prove that W = N (T2) is an invariant subspace for T, and that the restriction TW of T to W satisfies (Tw)2 = 0. If; we NHL) , 4.4% Tltww = T3M =T(rlcvn=TLa)=o. ”Ms demos TMéNLT‘), so )JCTZ) :: )v‘vm‘mk ’6») cums” at Tw, {l WNW), 1km (razor TwlTu/(UN = ”mm = fluke. $9 (-5020. 4. Let ka denote the subspace of PAR) consisting of polynomials f (x) such that f(1), f(2), . . . ,f(k) are all equal to zero. Find dim(Wn,k), in terms of n and k. You will probably need to consider the cases k; S n 4“ and k 2 n + 1 separately. 1-? land) M £0) Ewfi'k ivah'es 3C :5 a. ”Lamb/‘4 gear-u. an wi‘lk >\A mots, Ma; 3&0, TWV‘QM WM“ :0) @va— S9 O‘RMkal =0, in Ms we. 15 km, m we, “1’ :2: min—up“ tartar ‘IAWYOlLHc/w. SD WCE) :lfl, Maw ka 1 NCE) ’ Mm 0% (wk) : qu‘r», (63 = dim (Pmmn- Wfi) = wet-W». 'm Jflw‘fi 0152- . 5. (a) Find the real 3 X 3 matrix M such that M has eigenvectors 1 0 1 vi: 1 , v2: 1 , and v3: 1 7 1 1 0 with corresponding eigenvalues A1 = 0, A2 = 1, A3 = —1. 'Pvui Q1<|\l> M C&\¢w\w\€ Q :(‘OHOUI \ i 0 \ ~\ ' -\ ~\ \ Tm m=Q(2‘C§)Q :60” 00'\ ~\ to ’ Sims». CM CM Emu—LR) Chad; :3“ 0,034,)” \Atj (14¢.quwa MU” Mvs, wall Mvs, we wik moi \M. “W ope/«MS wik owls-<62 crepe/ii Jar mfihm‘fic exam who; 6. For which real numbers t does the system of linear equations ( i) (E) (i) have a non-zero solution x? Justify your answer. \m m& ion Wilcox A «A mCRaohis in \9Q sw’ ‘24., oMiA) =0. EXVO'MQM? ‘93 “i“UL‘” am fast mm W “(mm 3&5 dAUH='Li-%Z+M\=%S, So £=s Awoiur £4)an in Soive. 1M Yrvmzw is is Rik/1M Wk(/-\\<3. Smog M 15“ ’2. was me. \«AQufiQO‘r/ ”Ms W; we 142.32 M W W iv ‘05 a Wm Wbtmi-im 0(n\)+btvrw. awrfiws was is as 1M3\«Mi§v’si— in: e/w‘oRQS, we. Cm} a=~\, \911, so “EM Min/Q W W91 he (\3s), L2.E=S r—lr—dr—a OOMH 7. Either diagonalize the matrix or ShOW that B is not diagonalizable. "TM (WOMSHC 7&5de ts dfl<fib 2: :‘$\ = £20.45} - ~q o 2—; that has. dgMVC/Q—kkeg out R=Z. Lu‘\"\\ WLHruCI‘b L M Aw, wfik WM‘H‘CUC‘E 2- Fm" eta—Smuhta‘m‘h‘hj‘ wt «20% m 1:0 otgswsgauz, Le. NOS), +0 have Mmsem 2, Bye,- W 1% mo W34 3} we \wMW‘fi 50 ka6)22_ > W92 ka€>§=l M aw menu. Tu; scams M4. e is WA— iémamUukk. 8. Use the Cayley—Hamilton Theorem to find real numbers a, b, c such that A"1 = a1 + bA + CA2, Where 1 2 —5 A = 0 3 4 . 0 0 —2 Tu chAm‘sln‘c WMM ‘-s P<t3=(«~m;~m—z~a = “L3+Z€7“+S(:~(p. 131 Ck5&3”HMH-m\, -/‘\3 +ZAL+SA“61‘O) Wu «lA3+JA+»A*—1 9. Using the standard inner product on R3 in which the standard basis {61,62,63} is orthonormal, find the orthogonal projection of the vector 61 =2 (1,0,0) on the subspace W: Span({(1 1,1), (1, 2,3)}). 76. Wt = 0")” <x.,x>= 3 all. w?”— <wa,>‘i>x 2 (“2.3) —- .SEOAJO :1 (4,0,1) <Xz,¥z>=2. 10. Let V be a finite—dimensional inner product space, and let {111, . . . ,Un} be an orthonormal basis of V. Let T: V —> V be a linear operator such that T*T = I, where T* denotes the adjoint of T. Prove that {T(vl), . . . ,T(vn)} is an orthonormal basis of V. We. W <TCV:>,TLV3>) 7- (vi, T*TCUQ) : (DchSW : l {I 12) O phi-£10151, TM; {QMIUS like} §T(v\)’...,'\‘(umfl is Cw w’hxovwsthQ 92*) Ma; «93‘1quer \ACQLKW‘b/ W01 0x b053, Shaw OUMCV\)'-‘Vl. (or, S‘MCQ T*T :1, T l; lwwrlilgb; qu WQM 5LT(u_\3,.-,T(vM)) 15 Q, bk§$.> ...
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