Econ 204
Supplement to Section 3.3
The Matrix Representation of a
Linear Transformation
The purpose of this note is to provide a brief treatment of quotient vector
spaces, and amplify on the relationship between linear transformations and
their matrix representations.
DeFnition 1
Given a vector space
X
and a vector subspace
W
⊆
X
, deFne
an equivalence relation
∼
by
x
∼
y
⇔
x
−
y
∈
W
Exercise 2
Show that
∼
is an equivalence relation.
DeFnition 3
We deFne a new vector space
V/W
,the
quotient
of
V
W
.
The set of vectors in
is
{
[
x
]:
x
∈
X
}
where we recall that [
x
] is the equivalence class of
x
with respect to the
equivalence relation
∼
. In other words,
[
x
]=
{
y
∈
X
:
x
−
y
∈
W
}
=
{
x
+
w
:
w
∈
W
}
Thus, each of the vectors is a
set
; this is a little weird at Frst, but try to
get used to it. Now, we have to deFne the operations of vector addition and
scalar multiplication. The deFnitions are
[
x
]+[
y
]=[
x
+
y
]
α
[
x
αx
]
One needs to check that these deFnitions make sense. [
x
] is a set, and there
are potentially many di±erent representatives, i.e. many
x
0
such that [
x
[
x
0
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 Fall '08
 ANDERSON
 Vector Space, CRD, Matrix representation, De La Fuente

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