University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Term Test – MATA22H – Linear Algebra I for Mathematical Sciences
Examiners: Sophie Chrysostomou
Mohammad N. Ivaki
Date: Wednesday, February 28, 2018
Duration: 110 minutes
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
SIGNATURE:
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO.
CIRCLE YOUR TUTORIAL:
TUT0001
HU Yuxuan
TU 16:00-18:00
IC 220
TUT0002
AGRAWAL Pankaj
FR 9:00- 11:00
IC 302
TUT0003
HU Yuxuan
TH 17:00-19:00
SW 128
TUT0004
ZHANG Xincheng
MO 8:00-10:00
HW 308
TUT0005
LOK CHAN Antonio
MO 15:00-17:00
HW 402
TUT0006
TEJADA Dominador
MO 13:00-15:00
IC 120
TUT0007
MO Zecheng
TU 18:00-20:00
IC 320
TUT0008
GILCHRIST Valerie
TH 17:00-19:00
IC 220
TUT0009
LI Jinglin
TU 18:00-20:00
IC 200
TUT0010
YANG Xinling
TH 8:00-10:00
MW 262
TUT0011
LE Uyen
MO 9:00-11:00
HW 408
TUT0012
WEI Xuan
FR 13:00-15:00
HW 308
TUT0013
LIN Zihao
WE 19:00-21:00
BV 361
TUT0014
HUANG Wenyu
FR 9:00-11:00
AA 206
TUT0015
ZHANG Jingxuan
MO 9:00-11:00
MW 110
TUT0016
ZHAO Jinman
TU 20:00-22:00
IC 200
TUT0017
AN Qiang
TH 9:00-11:00
MW 110
TUT0018
LIN Zihao
WE 8:00-10:00
MW 223
TUT0019
YANG Xinling
MO 15:00-17:00
IC 208
TUT0020
LE Uyen
FR 9:00-11:00
SW 143
TUT0021
TEJADA Dominador
MO 11:00-13:00
MW 140
TUT0022
CHAN Bryan
TH 9:00-11:00
PO 101
TUT0023
LI Jinze
FR 10:00-12:00
HW 308
TUT0024
LI Tangling
FR 9:00-11:00
MW 120
TUT0025
ZHANG Xincheng
TH 18:00-20:00
BV 361
TUT0026
XU Yuqing
TU 15:00-17:00
HW 308
NOTES:
• No calculators, or any electronic aid is permitted. If you have a cell phone in the room,
make sure it is away from you and turned off.
• No books, notebooks or scrap paper are permitted.
• There are 14 numbered pages in the test. It is your responsibility to ensure at the start
of the test that this booklet has all its pages. Pages 13 and 14 are empty.
• Please leave all the pages of this booklet stapled. Do not remove any pages.
• Answer all questions in the space provided. Show your work and justify your answers
for full credit.
FOR MARKERS ONLY
Question Marks
1 /9
2 /9
3 /8
4 /8
5 /9
6 /8
7 /8
8 /8
9 /9
10 /8
11 /8
12 /4
13 /4
TOTAL /100
MATA22H3 page 1
1. (a) [3 points] Find all real values c such that the vectors u = [3c2 – 2, c3 – 2, c4 – 6]and v = [10, -10, 10] are parallel.
(b) [3 points] Find the orthogonal projection of a = [1, √6, 1] on u = [4, 0, -2].
(c) [3 points] Let A = 1 5 1 7 2 8 – –24 2 . Describe all vectors b = b b b1 2 3 such that
the system Ax = b is consistent.
MATA22H3 page 2
2. [9 points] Let the points (5, -1, 2), (7, 0, -1), (9, 6, 7) be vertices of a triangle in R3.
Determine the length of each of the three sides of the triangle. Also, determine the
three interior angles of the triangle (in the form of θ = arccos(x)).
MATA22H3 page 3
3. [8 points] Find the shortest distance from the point (0, 2, -3) to the line that goes
through the points (1, -1, -2), (2, -2, -2).
MATA22H3 page 4
4. [8 points] Let ABCD be a trapezoid with sides AB and CD parallel. Let M1 and
M2 be the midpoints of the nonparallel sides (AD and BC). Use vector methods
to show that the vector —-→ M1M2 = 1
2 -→ AB + –→ DC. Hint: express 2—-→ M1M2 in terms of
–→
AD, -→ AB, –→ BC, and –→ DC.
MATA22H3 page 5
5. [9 points] (a) State the Cauchy- Schwarz inequality.
(b) Give the definition of the span of vectors v1, v2, · · · , vm ∈ Rn.
(c) Give the definition of an elementary matrix.
| MATA22H3 | page 6 |
| 6. [8 points] Suppose c1, c2, c3, c4, c5 ∈ R5 and they satisfy the following two equations: | |
| 2c1 + c2 + -c1 + 2c2 – c3 |
– 3c4 – c5 = 0 = 0 |
AssignmentTutorOnline
Furthermore it is given that the vectors c1, c2, c4 are linearly independent.
If A is the matrix that has as its ith column the vector ci for i = 1, 2, 3, 4, 5, then find
the row reduced echelon form of A. Justify your answer.
MATA22H3 page 7
7. (i) [4 points] Let A be an m × n matrix and B be an n × k matrix. Prove that
(AB)T = BTAT .
(ii) [4 points] Let A be an m × n matrix, and let c be a column vector such that
Ax = c has a unique solution.
a. Prove that m ≥ n.
b. If m = n, must the system Ax = b be consistent for every choice of b?
c. Answer part (b) for the case where m > n.
MATA22H3 page 8
8. (i) [3 points] If AT = -A, then we call A a skew-symmetric matrix. Suppose A and
B are both skew-symmetric matrices of the same size and r, s ∈ R. Prove that rA+sB
is a skew-symmetric matrix.
(ii) [5 points] Suppose that A is an m×n matrix and B is an n×n invertible matrix.
Prove that the column space of A, C(A), is equal to the column space of AB, that is
C(A) = C(AB).
MATA22H3 page 9
9. [9 points] Let A = -11 2 0 3 5 2 1 -1 1 -3 0 4 -2 -4 and b = – -7 97
(a) [6 points] Use the Gauss-Jordan method to find the general solution to Ax = b.
(b) [3 points] Give the nullspace of A.
MATA22H3 page 10
10. [8 points](a) [3 points] Give the definition of a subspace W of Rn.
(b) [5 points] Suppose that W is a subspace of Rn. Prove that
W⊥ = x ∈ Rn x · w = 0 for all w ∈ W
is also a subspace of Rn.
MATA22H3 page 11
11. [8 points] Let A =
| -1 1 1 0 | 0 1 |
3 1 -1 .
Find A-1 and express it as a product of elementary matrices.
MATA22H3 page 12
12. [4 points] If A-1 =
| – 2 -3 1 | |
| -1 | 0 2 |
| 1 | 2 1 |
and B = 2(A)T , find B-1.
13. [4 points] Find all complex numbers z satisfying z2 = i.
MATA22H3 page 13
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MATA22H3 page 14
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