Also, Eq. (11.37) can formally be written as
A x
ð Þ ¼ e 1 Á Á Áe n
ð
ÞA
x 1
⋮
x n
0
B
@
1
C
A ¼ e 1 AÁ Á Áe n A
ð
Þ
x 1
⋮
x n
0
B
@
1
C
A,
ð11:40Þ
where we assumed that the distributive law holds with operation of A on (e 1 Á Á Áe n ).
Meanwhile, if in (11.37) we put x i ¼ 1, x j ¼ 0 ( j 6 ¼ i), from (11.39) we get
A e i
ð Þ ¼
X n
k¼1
e k a ki :
Therefore, (11.39) can be rewritten as
A x
ð Þ ¼ A e 1
ð Þ Á Á Á A e n
ð Þ
ð
Þ
x 1
⋮
x n
0
B
@
1
C
A:
ð11:41Þ
Since x i (1 i n) can arbitrarily be chosen, comparing (11.40) and (11.41) we
have
e i A ¼ A e i
ð Þ 1 i n
ð
Þ :
ð11:42Þ
The matrix representation is unique in reference to the same basis vectors.
Suppose that there is another matrix representation of the transformation A such that
A x
ð Þ ¼ e 1 Á Á Áe n
ð
Þ
a
0
11
Á Á Á a
0
1n
⋮ ⋱ ⋮
a
0
n1 Á Á Á a
0
nn
0
B
@
1
C
A
x 1
⋮
x n
0
B
@
1
C
A:
ð11:43Þ
Subtracting (11.43) from (11.37), we obtain
e 1 Á Á Áe n
ð
Þ
a 11 Á Á Á a 1n
⋮ ⋱ ⋮
a n1 Á Á Á a nn
0
B
@
1
C
A
x 1
⋮
x n
0
B
@
1
C
A À
a
0
11
Á Á Á a
0
1n
⋮ ⋱ ⋮
a
0
n1 Á Á Á a
0
nn
0
B
@
1
C
A
x 1
⋮
x n
0
B
@
1
C
A
2
6
4
3
7
5
¼ e 1 Á Á Áe n
ð
Þ
∑
n
k¼1 a 1k À a
0
1k
À
Á
x k
⋮
∑
n
k¼1 a nk À a
0
nk
À
Á
x k
0
B
@
1
C
A ¼ 0:
442
11 Vectors and Their Transformation