Ay¸
se Alaca
MATH 2107
(Eigenvectors & Linear Transformations)
3
The Matrix of a Linear Transformation
We willl look at the matrix factorization
A
=
PDP

1
in terms of linear transforma
tions.
Let
V
be an
n
dimensional vector space,
W
be an
m
dimensional vector space,
B
be a basis for
V
, and
C
be a basis for
W
.
Let
T
:
V
→
W
be a linear transformation.
We know that for any
v
∈
V
, [
v
]
B
∈
R
n
and [
T
(
v
)]
C
∈
R
m
.
Question:
What is the connection between [
v
]
B
and [
T
(
v
)]
C
?
Answer:
Let
B
=
{
b
1
, b
2
, . . . , b
n
}
. Let
v
∈
V
.
Then there exists scalars
r
1
, r
2
, . . . , r
n
such that
v
=
r
1
b
1
+
r
2
b
2
+
· · ·
+
r
n
b
n
=
⇒
[
v
]
B
=
r
1
r
2
.
.
.
r
n
.
T
(
v
) =
T
(
r
1
b
1
+
r
2
b
2
+
· · ·
+
r
n
b
n
) =
r
1
T
(
b
1
) +
r
2
T
(
b
2
) +
· · ·
+
r
n
T
(
b
n
)
[
T
(
v
)]
C
= [
r
1
T
(
b
1
) +
r
2
T
(
b
2
) +
· · ·
+
r
n
T
(
b
n
)]
C
=
r
1
[
T
(
b
1
)]
C
+
r
2
[
T
(
b
2
)]
C
+
· · ·
+
r
n
[
T
(
b
n
)]
C
.
Since [
T
(
b
i
)]
C
∈
R
m
(1
≤
i
≤
n
), we can write the above vector equation as a matrix
equation
[
T
(
v
)]
C
=
h
[
T
(
b
1
)]
C
[
T
(
b
2
)]
C
. . .
[
T
(
b
n
)]
C
i

{z
}
[
T
]
C←B
r
1
r
2
.
.
.
r
n
=
[
T
]
C←B
[
v
]
B
(1)
The matrix
[
T
]
C←B
is a matrix representation of
T
, and it is called the
matrix for
T
relative to the bases
B
and
C
.