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exam2_answers - Linear Algebra I MAS4105(6137 ‘ Second...

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Unformatted text preview: Linear Algebra I: MAS4105 (6137) ‘ Second Hour Exam W. Mitchell 1 Wednesday November 3, 2010 Start your answers on the worksheet; if you need more space, 1. (15 ) continue them on the back. Be sure to show your work to receive 2' (20 ) _— full credit. _ 3 15 —— You may use an 8 1/2 by 11 one—sided crib sheet as long as you ’ ( ) do not write too small. No calculators are allowed on this test. 4 (15 ) 5. (15 ) 6. (20 ) Tot. (100) 1. (15 points) For each of the following questions, circle either True or False. (a) AB 2 I implies that A and B are invertible. True or (b) Let T be a linear operator on a ﬁnite—dimension vector space . Then for any ordered bases ,8 and 'y for V, [The is similar to [T]7. w or False (c) The product of two matrices always has rank equal to the lesser of the ranks of the two matrices. , ’ True or - 8/, Ada (3;). (d) A system of 11 linear equations in n unknowns has at most one solution. True or- (e) If M E Mnxn(7€) and c E R then det(cA) = cdet(A). True or .‘ﬁg (100 (c/V -‘ c ” MM) (f) There exists a square matrix with no eigenvalues. or False o l 1 a; (to o M MIX/R). (g) There exists a linear operator T with no T—invariant subspace. True or V r [a] Mr W} Far/m/‘MZ‘ 2. (20 points) For each of the following matrices, state whether it is diagonalizable and, if it is, ﬁnd Q and D so that A = QDQ‘1 where D is diagonal. (Do not ﬁnd Q‘l. (a) A:(0 0) ye}! ‘D=4) 6?:1'. 3. (15 points) Prove the following theorem: Theorem. Suppose that T is a linear2 operator on the vector space V, and W is a T-z’nvam’ant subspace of V. Then W is Tn-invariant for every positive integer n. F’VVf M W /2c/a&I//‘W M ’7’ 2‘" MI, MWW%7;7 4/ Ma 73/» Mir/M & y/ﬂfé/c 7%» 47/2 3/ AMI///m/ M/gm ﬂﬂﬂv /A ﬂV/m /97ZIJ 747/ W/j fﬂa/Iq rg/IM mﬁrp H66 76r)éA/é/iz'éaé~c/‘ 9/;1/44’” 11/- 70/ “7(7”[x//EW 5/MW‘W/ 774th f I ' /p1e’ X W 4////fﬂ//’ WA; 7””: ,’;7/4,/M re ﬂag”? “ﬂag/rownoéeﬂz, W W/l f}, /;M/ﬂ/ﬂﬂﬂ’ [j 4. (15 points) Find all solutions to the following system of linear equations: \$1+2\$2+\$3+\$4=1 mgfx3+x4=1 I2 / / / , (0 , -, ,/, .. a-l .., K’ 0 3 ’/ ) 0 /—// / )0 +%>9«J<,=~/ 5*” X15"/ “3/5 ”W XLr/ +X3 a; 5. (15 points) Prove the following theorem: Theorem. For any square matrix A, the transpose At has the same characteristic polynomial as A, and hence the same eigenvalues. 64¢ M”) = yéf (Ah—AI) .— WMZ’AIL‘) : Wf/A‘aﬁf) : M(/4"/\I) .—.— 04/64) 13 6. (20 points) Below are three bases for P2 (R). 5 : {1, \$5 \$2} ”Y = {932,151} (5 = {1+\$,1,\$+a:2}. Find ﬁg]:— 0 & (a) The matrix for converting from 'y to ,8. (0 ‘ . l 0 . . I I (b) The matrlx for convertmg from 6 to ﬂ. { 0 0 0 (c) The matrix for converting from 3 to 6. 0 l / / O (7 m \/ /\ O‘x“ ng “o \/ \‘ ...
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