Lecture 1m
The Change of Coordinates Matrix
(pages 221-224)
At the end of the previous lecture, we found the coordinates of p(x) = 62x+2x2
with respect to two different bases. It happens that sometimes you will start
with the coordinates for a vector with

Assignment 1
[1pt] 1. When are assignments due?
Assignments are due every week at 11:55pm Eastern Time on the date indicated
on the course schedule. (Usually on Sunday.)
[1pt] 2. What file formats are acceptable for assignments?
Assignments must be submit

Assignment 5
7
5
3
[2pt] 1. Is the set 1 , 4 , 10 orthogonal?
5
5
1
To determine if the set is orthogonal, we need to see if each pair of vectors in
the set is orthogonal:
3
7
1 4 = 21 + 4 25 = 0
5
5
3
5
1 10 = 15 + 10 5 = 0
5
1
7
5
4 10 =

Assignment 9
[2pt] 1. Use polar form to calculate (2 2i)6 .
First we need to find the polar form of 2 2i, and the first step in that process
is to compute
r=
p
22 + 22 = 2 2
Now we
need to
find . We know that cos = 2/(2 2) = 2/2, and sin =
2/(2 2) = 2/2

Assignment 11
[2pt] 1. When is your final exam?
Actually, I dont know when your exam is, as the schedule can be different for
every student. So please make sure you double and triple check QUEST so that
you know when and where to be.
2
[8pt] 2. Let A = 1

Winter 2016
MATH 225 Online
University of Waterloo
Course Schedule
IMPORTANT: ALL TIMES EASTERN - Please see theUniversity Policies section
of your Syllabus for details.
Module
Week
Readings
Assignments
Due Date
Weight (%)
Module 01: Vector
Spaces
Week 01

Lecture 1h
Bases
(pages 206-209)
The next step after defining a spanning set and linear independence is to look
at a basisa set that is both at the same time! We did not pay much attention
to bases in Math 106, but they will play a much greater role in th

Assignment 3
[2pt] 1. Find the coordinates of p(x) = 6 + x 3x2 with respect to the basis
B = cfw_1 + x2 , 1 x + 2x2 , 1 x + x2 of P2 .
This means we need to find scalars a, b, c R such that
a(1 + x2 ) + b(1 x + 2x2 ) + c(1 x + x2 ) = 6 + x 3x2
This is th

Assignment 2
[4pt] 1. Prove that the set
A = cfw_a0 + a1 x + a2 x2 + a3 x3 | 2a0 + a2 = 0, a1 + 4a3 = 0, a0 , a1 , a2 , a3 R
is a subspace of P3 .
S(0): Since 2(0) + 0 = 0 and 0 + 4(0) = 0, we see that 0 = 0 + 0x + 0x2 + 0x3
is in A.
S(1): Let a(x) = a0 +

MATH 225 Spring 2016: Assignment 1
I.D. Number:
First Name:
Due at 8:25 a.m. on Friday, May 13, 2016
Family Name:
List any references that you used beyond the course text and lectures (e.g. discussions, texts or online
resources). If you did not use any a

Assignment 4
[3pt]
1. Determine
the matrix of the linear mapping L : M (2, 2) P3 defined
a b
by L
= b + cx + dx2 + ax3 with respect to the basis
c
d
1 0
0 1
1 1
0 1
B=
,
,
,
for M (2, 2) and the basis
0 1
1 0
0 0
0 1
C = cfw_1 + x, 1 x, x2 + x3 ,

Assignment 6
1
[1pt] 1. Find a basis for the orthogonal complement of Span 2 .
5
1
The orthogonal complement of Span 2 is the set of all ~x such that
5
1
~x 2 = 0
5
Computing the dot product, this is the same as finding the solution to the
system
x1 +

Lecture 2d
A Note on Rotation Transformations
(pages 329-330)
Back in Chapter 3, the textbook briefly discusses the matrix of a rotation around
a coordinate axis in ]R3.At that time, the text book also noted that we would
not be able to find the matrix of

Lecture 1s
Isomorphisms of Vector Spaces
(pages 246-249)
Definition: L is said to be one-to-one if L(u1 ) = L(u2 ) implies u1 = u2 .
Example: The mapping L : R4 R2 defined by L(a, b, c, d) = (a, d) is not oneto-one. One counterexample is that L(1, 2, 1, 2

Lecture 2a
Orthonormal Bases
(pages 321-323)
Lets stop and think about the features we really want in a basis. By its definition, it is a linearly independent spanning set, but the reason we wanted those
features is that we wanted to be able to uniquely w

Lecture 1n
Linear Mappings
(pages 227-228)
Lets return now to a concept we developed in our study of Rn linear mappings.
Because, as with bases, linear mappings will be of much greater use in the
general world of vector spaces than they were in simply the

Lecture 1j
Dimension
(pages 211-213)
Before we look at extending a linearly independent set into a basis, we will need
a few more facts. In our technique for shrinking a spanning set to a basis, we
depend on the fact that, if we get small enough, we will

Lecture 1g
Span and Linear Independence in Vector Spaces
(page 204)
One of the common ways to define a subspace is to think of it as the set of all
linear combinations of a set of vectors. First, lets note that this IS a subspace!
Theorem 4.2.2 If cfw_v1

Lecture 1q
The Matrix of a Linear Mapping
(pages 235-239)
When we first studied linear mappings in Math 106, they were only between
Rn and Rm . In this setting, we were always able to find a matrix A such that
our linear mapping L(~x) was the same as A~x.

Lecture 1l
Coordinates
(pages 218-221)
Once we have a basis B = cfw_v1 , . . . , vn for a vector space V, the Unique Representation Theorem tells us that for any given vector x V, there is a unique
way to write x as a linear combination of the vectors in

Lecture 1r
Change of Coordinates and Linear Mappings
(pages 240-242)
Lets take another look at an example from the previous lecture:
Example: Let L : M (2, 2) P2 be defined by
L
a
c
b
d
= (a + b) + (a + c)x + (a + d)x2
We found the matrix for L with respe

Lecture 1o
Range and Nullspace
(pages 228-230)
There are two subspaces that we affiliate with linear mappings: the range and
the nullspace. We will define them first, and then prove that they are subspaces.
Definition: Let V and W be vector spaces over R.

Lecture 2f
Projections Onto A Subspace
(pages 334-336)
Now that we have explored what it means to be orthogonal to a set, we can
return to our original question of how to make an orthonormal basis. We will
construct such a basis one vector at a time, so f