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Unformatted text preview: THE FACULTY OF ARTS AND SCIENCE
University of Toronto FINAL EXAMINATION, December 2004 5%
MAT 223H1 F  Linear Algebra I vs)?" Examiners: S. Cohen
H. Li
P. Milman
R. Moraru
R. Stanczak Last Name: Student # First Name: (a) TIME ALLOWED: 3hours
(b) NO AIDS ALLOWED
(0) TOTAL MARKS: 100 (d) WRITE SOLUTIONS IN THE SPACE PROVIDED (e) GIVE ANSWERS FOR THE MULTIPLE CHOICE QUESTIONS ON
THE SECOND PAGE, ONLY THOSE ANSWERS WILL COUNT (f) DO NOT REMOVE ANY PAGES. THERE ARE 16 PAGES
(9) MARKING SCHEME: (1) Multiple Choice Questions : see page 3 (2) Written Questions: indicated by each question MARKER'S REPORT ns
ns ns _______.______.————— MC /48
TOTAL I100 NAME: ' Student No Give answers for multiple choice questions below.
ONLY THOSE ANSWERS WILL COUNT Right way of giving the answer : Q PART I — MULTIPLE CHOICE QUESTIONS
MARKING SCHEME: 4 marks for each correct answer 3 ] and let ()CA)"l = AT. Find the entry in the second row, second 1
1. Let A=[ column of the 2x2 matrix X. @@@@®®
(J1 2. Which of the following is a subspace of R3 ? A. W={(x,y,z)x=z}
13 W={(X,y,Z)IXyZ=0}
C. W= {(x,y,z)lx+y=1} all A and C
B and C
A only
B only @@®@®® none 3. For what value of k is the set S = {x2 — x, x —1, x2 + kx + 1 } a basis for P2 ? ® @@@@® k¢—2
k¢2
k¢~1
forall k
k¢0 there is no such a k 4.Findthevectorin W=span{(1,—1,0),(1,1,0)}closestto X=(1,0,—1) 6) (1,0,0)
(2 (2,0,0)
(3 1/2(1,0,—1)
<4) (0,0,0)
(:9 (1,—1 ,0)
6) (2,1,1) 5. If the GramSchmidt orthogonalization process is applied to a basis
B={(1,—1,0,1),(1,1,0,0),(1,1,0,1)}
for the subspace W of R4 , then the orthogonal basis obtained is
Bo={(1,—1,0,1),(1,1,0,0),X}.Find X. 6) %(—1,0,0, 1) ® %(—1,2,0,~1)
© (—1,—1,0,0)
@ —;(—1,1,0,2)
© 1
© ——1,1,2,2
3( ) (0, 0, 0, 0) R1
R7 _ .
6. Suppose A = R‘ ,where Ri denotes the z—th row of A. Flndanelementary
3
R4
R1
. R 3R
matr1x E such that EA = 2; 3 for all A.
3
R4
1 O 0 0 l 0 0 O l 0 0 0
0 l 0 0 0 1 3 0 0 1 0 3
G) (2) ®
0 0 3 0 O 0 l 0 0 0 1 0
0 0 0 1 0 0 0 l 0 0 O l
1 0 0 0 l O 0 O 1 0 0 0
0 l 3 0 0 3 3 0 0 3 l 0
® 6) ©
0 0 3 0 0 0 l 0 0 O 1 0
0 0 0 1 O O 0 1 0 0 0 l 7. Suppose that the system of linear equations represented by the augmented matrix a b c — 2b
d e f — 2e has a unique solution (x1 , xz , x3 ). Find X2 .
g h k — 2h 6) 0 ® —1 (3D 2 @ % C5) cannot be determined from the given data © ~2 8. Suppose that none of the eigenvalues of an n><n matrix A are equal to zero. Which of the following statements must be true?
A. A is similar to A".
1 + A B. If A isaneigenvalue of A then [I is an eigenvalue of A”1 + 1,, . C. A is diagonalizable. all A and C
B and C
A only
B only @@@®®® none , i.
006$ . 0 2k 0
9. Let A = 0 2 . For what value(s) of k detA = 0 ?
k + 1 k —1 1 0 k—l 1—2k 0
k=l only
yk=0 only
k=lor—2
k=0,lor2
k=0,lor—2
k=lor2 @@®@®® 10. Suppose A is a 3x3 invertible matrix and a matrix B has three columns. Which of
the following is always true? A. If AB is invertible, then B is invertible.
B. If AB= —(AB)T, then B is not invertible.
C. If AB=BA,then B is invertible. all B only
C only
A and B
B and C none @@@@®® coded“? 11. Let T: R7 —> R8 be deﬁned by T(X) = AX. Which of the following is always equal
to rank of A?
A. 7 — dim(kerT) B. 8 — dim(imT)
c. 7 — dim[(rowA)L] (D A only
(Z all
(3 C only
(4) none
® B and C
G) A and C 1
12. Let T: R3 —> n3 be deﬁned by T(X) = 2X — (XN)N, where N = 0 . Find the third column of standard matrix representation of T. —1 1 l
0 0 ® 1
1 —l 0 1
“l
0 (D Q) Q!) ...
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 Spring '09
 UPPAL
 Linear Algebra, Algebra, P. Milman

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