2003 - 2004 Spring Final with key - b c b-c d 2c-d a a...

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MATH 260, Linear Algebra Final Exam May 31 , 2004 1.(25 pts) Consider the matrix A = [ -i a+2 For which value of a E R is A (i) of rank I? (ii) of rank 2? (Hi) of rank 3? (iv) of rank 4? 1 2 0 ] a+I 3 a-I a-2 a-5 a+I 2 a + 4 -2a Noh Hut JrJ A :: - ,,1. (4 + 1) . t-J.,tt 4\SO thAt f~c. firtt ~W'o roW', ff A Arc Alw.Js liW(4Y~ ItH~ept"dlnt. "-low it (4~ ~t r'kti"'e~ th,ck,,", tho\t (n A It A A ( it) (iii) (iv) is n (vt."" of (' A n k 1" -- --. ... - IS of r'A" 2 If t.\ -= 0 .. i.S of rAttk 3 If A :::-1 . is .f r.h k 1; if A::fO .",( Af-.
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2. (25 pts). (i) Find the eigenvalues and eigenvectors of [ 1 1 3 ] A= 0 2 1 4 -4 -1 It is known that 2 is an eigenvalue of A. (ii) Is A diagonalizable? If yes find a matrix B such that B-1 AB is diagonal. Otherwise explain why such a matrix does not exist. 1 ' .! ./1 0 - \ A i$ ai,,~.n~\'S4\\c. 4rlA 2.. .. I ~ 1 \ -~ Iv. ~=~ ;-~ ,- 1 J I r1 '2. 1- 1- 1 I t;:: 1 1 -1 2- ;:! ~ I t () 1 -10 "'10
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3.(25 pts) Let A and B be two matrices in Rnxn. Assume that A, B, A + Bare nonsingular and (A + B)-I = A -I - Put C = AB-I (i) Prove that C + C-I = -1 (ii)Prove that C3 = 1 (i) (ii) . C -I , +- (, "If A~- ~ IbA-' :: A I (Ai- 1>5' - A-'J + I) lrA+ ~)'- ~,] ( . I ..., -I 1 ::: tA ;- ~) ( A r ~ ) - A A - ~ " ::: - . C'} :: ~ C-11'): :: (C - I) ( C 1k (+-1) + I ::: (c - I ) ( ( C ~ :r ~ t') + 1 ~ 0 ::. :l
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4.(25 pts) Let T(a, b, c, d) = (a +
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Unformatted text preview: b + c, b-c + d, 2c -d + a) a linear transformation from R4 into R3; (i) Show that the set W = {(a,b,c,d) E R4: T(a,b,c,d) = (O,O,O)} is a subspace of R4. (ii) Represent T with respect to the bases B = {VI = (0,1,0,0), V2 = (1,0,0,0), V3 = (0,0,0,/;, V4 = (0,0,0,1) andC= {WI= ((1,1,0),w2 = (1,0,0)"w3 = (0,0,I)} , \ . (\) For-..~ + ht. sc.t-Ktr' 'f . U .S'"hPAC' Q 'intA-\'" +rAn.sfor-MAtiort 'f: L1 ~ V-i ~-. :: rr (0 ) ~ f-: f Vi 1 Cf (Lt ) = ~} 'F tA j iOl.4t ihe. . ( i i) C he c k fb ",r T-t--+ (rr.) ~ (1,1,0) :: W.-t-+--t--r ('\1'"1,) :: [ I, Ii) = W t.+ w, ~-1P ~ ~ I h'" ~) ;:: (1,-I, 2)-=-"'i + .2 W to-I-2 IV; T ,-+ ~ ~ ("". .) =-(0, 1,--i) :::: W"-\tV 2.-W, IJ. . The \"f\Atr{)c (tpr(Stn~l~ T Wtr.to rhL [-i-t-f--f ~ ~ } . b A.St s 1 ~ , ~ , l'f, I ~ 1 (\t\d f Wi J 'W Co I WJ I S 1-1 2-).. 1-1-1 1 1...
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2003 - 2004 Spring Final with key - b c b-c d 2c-d a a...

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