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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.
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CW1 \wv Mamba \wcﬂapemow‘i “Loniions 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.
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+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
gearu. 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) = wetW». '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
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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 nonzero 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 Wbtmiim
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
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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”HMHm\, /‘\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)
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 Fall '08
 GUREVITCH
 Linear Algebra, Algebra

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