midterm3_yellow - SoLUTiO NS _ NAME: SECTION: MATH 21-241,...

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Unformatted text preview: SoLUTiO NS _ NAME: SECTION: MATH 21-241, MATRIX AL GEBRA Spring 201 1 Midterm 3 Section B (11:30-12:20) Total: 100 points - You must show ALL work for full credit. I - If you have any questions during the exam, please raise your hand. -' N 0 calculators, phones, outside materials are allowed. -- -- -—- 1. (19 pts) Mark the statements in a)—d) True or False. a) (8 pts) if the eigenvalues of A are 1,1,2, then A is diagonalizable- (Give a reason it true, or a counterexample if false.) m: Ar: 1 1 ‘ o i ( 0 o 2 D l ‘ O 0.30 190(7)”: .0 o I E: 0 10H; :0 etbro 0 0 i O b) (5 pts) If the eigenvalue matrix forA is A, what is the eigenvalue matrix forA + 21? (Justify your answer.) 11 A is an agrarian/9141 ole Alan Ax=Xv Fri-(11: Ax+1x [At-2:5)x =Q\+a)x :v K+z is anawmfiws o1£ ((+21 =) ‘A +11“. E c) (3 pts) If v1 and v; are linearly independent eigenvectors, then they correspond to distinct eigenvalues. (Justify your answer.) talk: We‘ve. Lem Cam Pr! with at W figmva lam“ M Pnchgndmt filafinvemri. d) (3 pts) if V is orthogonal to Wand W is orthogonal to Z, then V is orthogonal to Z. (Here V, W and Z are subspaces. Justify your answer.) Elfin ii 2 =\l, —\Jm‘s l‘flcuomhg leads t9 V ofilzofiwnak +51 Hate which “is lmPOJhw. 2. (16‘pts) The foiiowing parts are not related. a) (5 pts) If Q is an orthogonal matrix, is Q2 also an orthogonal matrix? (Give a proof if true, or a counterexample if false. if you give a counterexample, show why it works.) 1%: (Wm mime-Q) QTEE‘? 5W "*3 I :9 at is smegma ‘ b) (3 pts) Starting with the matrix A, interchange row 1 and row 3 to produce B; then subtract the first row (of B) from the third to produce C. How is detC related to detA? Justify your answer. Mbtpw‘A‘ MQeMB >LH~J c) (8 pts) If P : PTP, show that P is a projection matrix- ['0 ET: (PTP7T = PTU’TUT = PT? e P z) Plide Cu) P =5? = ‘3" =P1 => Piafdmrmf \‘ P 75 Q ?Na~{d\bn me. 3. (20 pts) a) (13 pts) Express the Gram-Schmidt orthogonalization for a1, a; as A : QR: a1=[122]T,a2=[1 31f. ' b) (7 pts) With the same matrixA as in part a), and with b = [1 1 IT, useA = QR to solve the least—squares problem Ax = b. R£=©fb [a 511“ ’1/3 2/3 2/3 z 0 (Z ‘3 o w; Mr; ‘ 4. (25 pts) Diagonalize the following matrix (if possible). if a diagonaiization exists, find S, S“ and A. 4 0 —2 A = 2 5 4 0 0 5 E valuu {4 X 0 ’1 2) _ ,x g.)\ ; O ‘5“ l M [1‘ =0 (5 Dflu X L01 a o o 54‘ A g n.3,: 5. (20 pts) 61) (12 pts) Find the third Legendre poiynomial. it is a quadratic polynomial! x2 +ax + b that is orthogonaf to 1, and x over the intervai —1 S x s 1. i <4,x1+ax+lo> = b) (8 pts) Suppose P is a projection matrix onto the line through a. Why is the inner product of x with Py equal to the inner product of Px with y? I<X. Pg? 7: xTPij <Px.g>;(Px)T.j = x1795 = ...
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This note was uploaded on 09/08/2011 for the course MATH 21-241 taught by Professor Irina during the Spring '08 term at Carnegie Mellon.

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midterm3_yellow - SoLUTiO NS _ NAME: SECTION: MATH 21-241,...

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