The Change of Coordinates Matrix
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
[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
[2pt] 1. Is the set 1 , 4 , 10 orthogonal?
To determine if the set is orthogonal, we need to see if each pair of vectors in
the set is orthogonal:
1 4 = 21 + 4 25 = 0
1 10 = 15 + 10 5 = 0
4 10 =
[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
22 + 22 = 2 2
find . We know that cos = 2/(2 2) = 2/2, and sin =
2/(2 2) = 2/2
[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.
[8pt] 2. Let A = 1
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University of Waterloo
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Module 01: Vector
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
[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
[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
Due at 8:25 a.m. on Friday, May 13, 2016
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
the matrix of the linear mapping L : M (2, 2) P3 defined
= b + cx + dx2 + ax3 with respect to the basis
for M (2, 2) and the basis
C = cfw_1 + x, 1 x, x2 + x3 ,
[1pt] 1. Find a basis for the orthogonal complement of Span 2 .
The orthogonal complement of Span 2 is the set of all ~x such that
~x 2 = 0
Computing the dot product, this is the same as finding the solution to the
A Note on Rotation Transformations
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
Isomorphisms of Vector Spaces
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
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
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
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
Span and Linear Independence in Vector Spaces
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
The Matrix of a Linear Mapping
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.
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
Change of Coordinates and Linear Mappings
Lets take another look at an example from the previous lecture:
Example: Let L : M (2, 2) P2 be defined by
= (a + b) + (a + c)x + (a + d)x2
We found the matrix for L with respe
Range and Nullspace
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.
Projections Onto A Subspace
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