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

Winter 2014
What is a theorem?
1. Direct method
If A then B
Suppose A    find out B
2. Contrapositive method
If B then A
Suppose B    find out A
3. Proof by contradiction
Suppose A and B    find out cont
Western University (Ontario)  Also known as University of Western Ontario
Linear Algebra
MATH 1600B

Winter 2014
#Assignment #4: Recursion
#Will Xie (10189120)
#Write a program for to find path through a grid recurvively
grid = []
def readFile(file):
theFile = open(file, 'r')
for line in theFile: #for each line
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: wro
Western University (Ontario)  Also known as University of Western Ontario
MATH 1600B

Fall 2018
8.1
Recall
21hm
:
Defy
J
If
E
[
=
good
Fix
check
E
6
21
E
Example
Zmn
e
digit
a
fixed
m
vector
to
respect
with
E
5
"m
'
'
d
that
so
T=[
J
,d]e
21
code
is
vector
for
E
message
1=0
mod
doesn't
This
to
Western University (Ontario)  Also known as University of Western Ontario
MATH 1600B

Fall 2018
1.2
The
Deft
Is
it
Notes
.
J
Deft
)
1
J
(3)
I.
T
or
(
E
R3
In
J=[
+
for
not
vector
a
with
vectors
2
n
components
.
product )
4
=
T=[Yn]eR
negative
E.
,
Ez
vz
3)
)
number
a
cross
)
of
non
"
vector
a
a
Western University (Ontario)  Also known as University of Western Ontario
MATH 1600B

Fall 2018
Chapter
I
Geometry
The
1.1
Vectors
:
Deft
A
vector
is
A
point
a
line
directed
a
from
displacement
Vectors
of
Algebra
and
to
initial
H
.
point
a
point
tail
or
to
corresponding
segment
B
AT
denoted
Is
M
Western University (Ontario)  Also known as University of Western Ontario
MATH 1600B

Fall 2018
Properties
of
A
,B ,C
Let
(
Algebra
Matrix
3.2
be
(2)
(
(3)
( Additive
(4)
(
( same
Span ( Ai
,
.
c.
A
(5)
+
( Btc )
O=A
A
C A)
+
for
as
cfw_
=
=0
.
Distributive
#
d)
Cet
)
(
( A
c
+
(
(7)
( Associa
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
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
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. alwa
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 equ
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
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.
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
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 matr
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 FAL
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 F
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 FALS
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.
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 FALS
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
Western University (Ontario)  Also known as University of Western Ontario
MATH 1600B

Fall 2018
Lines
1.3
Defy
Planes
and
I
for
vector
direction
A
l
to
parallel
determine
Any
.
direction
a
of
l
line
a
PE
vector
Is
P
points
distinct
2
"
R
in
for
,Q
l
non
a

line
the
on
vector
zero
l
Exy
=0
A
eq