Example 11.2 Let e 1 , e 2 , e 3 , e 4 be basis vectors of V
4 . Let A be an endomorphism of
V
4 and described by
A ¼
1 0 1 0
0 1 1 0
1 0 1 0
0 1 1 0
0
B
B
B
@
1
C
C
C
A
:
We have
e 1 , e 2 , e 3 , e 4
ð
Þ
1 0 1 0
0 1 1 0
1 0 1 0
0 1 1 0
0
B
B
B
@
1
C
C
C
A
¼ e 1 þ e 3 , e 2 þ e 4 , e 1 þ e 2 þ e 3 þ e 4 , 0
ð
Þ :
That is,
A e 1
ð Þ ¼ e 1 þ e 3 , A e 2
ð Þ ¼ e 2 þ e 4 , A e 3
ð Þ ¼ e 1 þ e 2 þ e 3 þ e 4 ,
A e 4
ð Þ ¼ 0:
We have
A 2 e 1 À e 2 þ e 3
ð
Þ ¼ À A e 1
ð Þ À A e 2
ð Þ þ A e 3
ð Þ ¼ 0:
Then, we find
A V
4
À Á ¼ Span e 1 þ e 3 , e 2 þ e 4
f
g , Ker A ¼ Span 2 e 1 À e 2 þ e 3 , e 4
f
g : ð11:49Þ
For any x 2 V
n
, using scalar c i (1 i 4), we have
x ¼ c 1 e 1 þ c 2 e 2 þ c 3 e 3 þ c 2 e 4
¼
1
2
c 1 þ c 3
ð
Þe 1 þ e 3
ð
Þþ
1
2
2c 2 þ c 3 À c 1
ð
Þe 2 þ e 4
ð
Þ
þ
1
2
c 3 À c 1
ð
Þ 2 e 1 À e 2 þ e 3
ð
Þ þ
1
2
c 1 À 2c 2 À c 3 þ 2c 4
ð
Þ e 4 :
ð11:50Þ
Thus, x has been uniquely represented as (11.50) with respect to basis vectors
(e 1 + e 3 ), (e 2 + e 4 ), (2e 1 À e 2 + e 3 ), and e 4 . The linear independence of these vectors
can easily be checked by equating (11.50) with zero. We also confirm that
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11 Vectors and Their Transformation