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Unformatted text preview: SoLUTiO NS _ NAME: SECTION: MATH 21241, MATRIX AL GEBRA
Spring 201 1 Midterm 3
Section B (11:3012:20) 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 ‘
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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
[At2:5)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 mimeQ) 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 GramSchmidt 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 21241 taught by Professor Irina during the Spring '08 term at Carnegie Mellon.
 Spring '08
 IRINA
 Algebra

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