Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
Complex numbers
We dene a complex number to be an expression of the form a+bi, where a and b are real numbers,
and i is a special symbol. The set of all complex numbers is denoted by C, and an individual complex
number a + bi can be represented by the poi
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
30% '
MATH 1600B Linear Algebra
[2 marks]
Page 1 of 2 Quiz 8, 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 6. always
true) or FALSE (tie. not always true). Provide a short justi
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
r
up
MATH 1600B Linear Algebra
Name:
[2 marks]
1: .
f'_'_._~_
[2 marks]
Page 1 0f 2 Quiz 1, 20 minutes
ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 8. always
true) or FALSE (ale. not always tr
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
I 00%
[2 marks]
7,
[3 marks]
MATH 1600B Linear Algebra Page 1 of 2
Quiz 9, 20 minutes
Name: ID Number:
(Please Print)
1. Determine whether the given statement is TRUE (ale. always true) or FALSE (2'. e. not
always true). Provide a short justication fo
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
905
[2 marks]
[2 marks}
\
MATH 1600B Linear Algebra Page 1 of 2 Quiz; 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 5. always
true) or FALSE (tie. not always true). Provide a sho
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
Math 1600B Linear Algebra Page 1 of 2 Quiz 2, 20 minutes
Name: 1 ID Number:
(Please Print) \
1. For each statement below, determine whether the given statement is TRUE (that is, always
true), or FALSE (that is, always false). Provide a short justification
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
Se cii'b OOS
MATH 1600B Linear Algebra Page 1 of 2 Quiz 4, 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (ale. always
true) or FALSE (tie. not always true). Provide a short justicati
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
C') I.) 5
MATH 1600B Linear Algebra Page 1 of 2 Quiz 5, 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 6. always
true) or FALSE (ale. not always true). Provide a short justicati
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
005
[2 marks]
[2 marks]
MATH 160013 Linear Algebra
Page 1 of 2
Quiz 7, 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 6. always
true) or FALSE 8. not always true). Provide a s
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
O O 3' ,
'\,)\
MATH 1600B Linear Algebra Page 1 of 2 Qniz 6, 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine Whether the given statement is TRUE (2'. 3. always
true) or FALSE (ale. not always true). Provide a short justic
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
.J a V
[2 marks]
[2 marks]
MATH 1600B Linear Algebra Page 1 of 2
Quiz 8, 20 minutes
Name: ID Number:
 (Please Print)
1. For each statement below, determine whether the given statement is TRUE ( 15.8. always
true) or FALSE (ale. not always true). Pr
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
l. {J
[2 marks]
[4 marks]
MATH 1600B Linear Algebra
Page 1 of2 '
Quiz 9, 20 minutes
Name: ID Number:
(Please Print)
1. Determine whether the given statement is TRUE (216. always true) or FALSE (2'. e. not
always true). Provide a short justication for yo
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
[2 marks]
[2 marks]
MATH 1600B Linear Algebra Page 1 of 2
Quiz 5, 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine Whether the given statement is TRUE (2'. 9. always
true) or FALSE (ale. not always true). Provide a short
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
gecJva cow
MATH 1600B Linear Algebra Page 1 of 2 Quiz 4, 20 minutes
Name: ID Number:
(Please Print)
I. For each statement below, determine whether the given statement is TRUE (2'. 6. always _
true) or FALSE (tie. not always true). Provide a short just
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
Eigenvalues and eigenspaces
We have been studying the following problem: given an n n matrix A, are there real numbers
c such that there exists a nonzero vector X Rn with AX = cX? Since AX = cX is equivalent to
A cIn )X = O, we see that this problem can b
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
A Markov chain example
1
3
2
3
1
2
1
2
, so P is a 2 2 transition matrix. Since P has no zero entries, P is a regular
Let P =
transition matrix. The characteristic polynomial of P is
1
1
2
3
2
1
2
3
det(P I2 ) = det(
)
1
2
1
1 2
1 1
= ( )( ) = 2 ( + ) +
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
Suppose that we have a Markov process with three states, having the transition probabilities as
shown in the following diagram:
0.2
0
0.2
1
2
0.1
0.3
0.6 0.2
0.9
3
0.5
0.2 0.1 0.3
0.2 0 0.2
0.6 0.9 0.5
Then the transition matrix for the process is P =
. L
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
A Markov chain example
In this example, we consider a 3 3 transition matrix P , which turns out not to be
diagonalizable over R, but is diagonalizable over C, and the theory that we developed
1
4
1
4
1
2
for R is still applicable for C. Let P =
2
3
1
3
0
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
A diagonalization example
We shall nd the characteristic polynomial, eigenvalues, eigenspaces and a basis for each, and
nd an invertible matrix P such that P 1 AP is a diagonal matrix for the 3 3 matrix
A=
0 2 3
1 1 1
2 2 5
.
The characteristic polynomial
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
1. For each of the following statements, circle T if the statement is always true and F if it can be
false. Give a brief justication for your answer.
[2]
(a) Let u, v, and w be nonzero vectors in R3 . If u and v are each orthogonal to w, then 2u 3v
is ort
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
O C) (l
MATH 16008 Linear Algebra Page 1 of 2 Quiz 1, 20 minutes
Name: ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 3. always
true) or FALSE (ale. not always true). Provide a short justication
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
0 04'
Math 1600B Linear Algebra Page 1 of 2 Quiz 2, 20 minutes
Name: w ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (that is, always
true), or FALSE (that is, always false). Provide a short justicati
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
DO Lt
MATH 1600B Linear Algebra Page 1 of 2 Quiz 3, 20 minutes
Name: ID Number: 1
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 6. always
true) or FALSE (ale. not always true). Provide a short justication f
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
")0le
[2 marks]
[2 marks]
MATH 1600B Quiz 6, 20 minutes
Linear Algebra
Page 1 of 2
Name: ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (316. always
true) or FALSE (ale. not always true). Provide a
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
30W
MATH 1600B Linear Algebra Page 1 of 2 Quiz 7, 20 minutes
Name: _ ID Number:
(Please Print)
1. For each statement below, determine whether the given statement is TRUE (2'. 6. always
true) or FALSE (ale. not always true). Provide a short justication f
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Spring 2016
Linear Algebra 1600a Midterm Solutions
7:0010:00 pm
October 24, 2008
(14 pts) 1. For each of the following, circle T if the statement is always true and circle F if it can be false.
Do not guess: wrong answers will receive 2 marks.
(a)
(b)
(c)
(d)
(e)
(f