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Unformatted text preview: THE FACULTY OF ARTS AND SCIENCE
University of Toronto FINAL EXAMINATIONS, AUGUST 2005 . 4 MAT223H1Y Linear Algebra I
9 Examiner: J. Callaghan, J. Chan Duration: 3 hours NO AIDS ALLOWED. Total: 100 marks Family Name:
(Please Print) Given Name(s):
(Please Print) Please sign here: Student ID Number: You may not use calculators, cell phones, or PDAs during ' the exam. Partial credit will be given for partially correct
work. Please read through the entire test before starting,
and take note of how many points each question is worth.
Please put a box around your solutions so that the grader
may ﬁnd them easily. MC
Problem 13:
Problem 14: Page 1 of 10 Multiple Choice Problems
Circle the correct answer. 4 marks for each correct answer. 1. Which of the following is an orthogonal set?
(A) {(1,0,0), (0,0,1), (0,1,—1)}
(B) {(0,0,0), (1,0,0), (0,1,0)}
(C) {(1,0, 1), (1,0,—1), (0,1,0)}
(D) {(1,1,1), (1,0,—1), (0,1,0)}
(E) {(1,0,0), (1,0,1), (0,0,1)} 1 3 2
2. Finddet 2 1 1 .
i (0 0 3)
(A) —15
(B) 0
(C) 3
(D) 21 (E) none of the above 1
3. Let A = ( (2) 2 g > . What is the dimension of E2(A)?
0 0 1
(A) 0
(B) 1
(C) 2
(D) 3
(E) none of the above Page 2 of 10 4. Let M and N be n X n symmetric matrices. Which of the following is NOT a symmetric
matrix? m)M+N
m)MNM (Q MN+NM
(D)M2
@)M+N+MN 5. Find all values of k such that the matrix ( Friltd
Pr
lel
I—‘HH ) has rank 2. (A) kaél
(B) k=0
(C) k=——1
(D) k9é2
(E) k=lork=—1 6. Let v1 = (2, —1, 1), 222 = (—3, —1, 1) and v3 = (—1,3, —3). If «13 = (101 +bv2, ﬁnd a.
M)? @)—3
@)4 @)*2
@)4 1 2
7. Let A = (1 2 3) and B = ( 4 5 g ) . Which of the following is deﬁned? 7 8 9 (A) A + 33 (B) 2AT4B Kn ABT+3A (m M+A (E) BTA + 3A Page 3 of 10 8. Suppose that the augmented matrix of a system has been reduced to 1 O 0 1 1
0 1 l 0 1 . Which of the following describes the nature of 1the solution set
0 0 0 0 0 of the system? (A) inﬁnitely many solutions with 3 parameters
(B) inﬁnitely many solutions with 2 parameters
(C) inﬁnitely many solutions with 1 parameters
(D) unique solution (E) no solution 9. Which of the following is NOT an elementary matrix? 001
(A)( ) A
w
V O H C @ A Bi V
coco OOH ONDi oo:— I A U v
AAAA
OD—‘OCJ‘HOOF—‘O COCO
I—Ioo Hoo H0O I—‘OO O
VVVV Page 4 of 10 10. Let U = span{(1,0,1), (0,1,1)} and v = (1,1,1). Find projU 'u.
(A) (1,0,1) (B) (1,1,1) 11. Which of the following is NOT a subspace of R3?
(A) {(56, y, 2W = y}
(B) {($,y,z)$3 = 23}
(C) {0641,31332 + 2/2 = 0}
(D) {(96,3/,z)=v2  22 = 0}
(E) {(m,y,z)lw + y + z = 0} 12. Find a matrix A such that null A = col A. (A) (33)
(B) ((1,?)
(c) (13)
(D)(‘1’8) (E) none of the above Page 5 of 10 Open Ended Questions 13. [8] Suppose E is a symmetric matrix. Let U = im E. Prove that U J = null E. Page 6 of 10 14. Let A be an n x n matrix such that A2 = I.
a. [6] Let 1) E R". Show that (I + A)v 6 EM), and (I — A)v E E_1(A). b. [4] Given any vector 'v E R", write 1) as a sum of vectors in E1(A) and E_1(A).
Hence, conclude that the eigenvectors of A span R". c. [2] Prove that A is diagonalizable. Page 7 of 10 15. a. [6] Let A be an 4 x 4 matrix. Suppose detA = 2. Find det(A3 adj (AT)).
b. [10] Let B be a 3 x 3 invertible matrix. Prove that 33 + B aé 0. Page 8 of 10 16. [8] Let A be an n X n matrix. Prove that A is invertible if and only if the columns of
A are linearly independent. Page 9 of 10 17. [8] Let A = OMI—I ) and let T : R4 —) R3 be the transformation induced by Q‘HON 1
2
0
A, Le. T(X) = A X. Fin a basis of her T. Page 10 of 10 ...
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 Spring '09
 UPPAL
 Linear Algebra, Algebra

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