Page 71 - Advanced Linear Algebra
P. 71
Vector Spaces 55
d
is a basis for the vector space = d over . Hence,
d
dim²= ³ ~ dim²= ³
d
Proof. To see that cpx²³ spans = d over , let % b & = d . Then %Á & =
8
(
)
and so there exist real numbers and some of which may be for which
1 1
%b& ~ # b @ #
A
~ ~
1
~ ² # b # ³
~
1
~ ² b ³²# b ³
~
8
To see that cpx²³ is linearly independent, if
1
² b ³²# b ³ ~ b
~
then the previous computations show that
1 1
# ~ and # ~
~ ~
8
The independence of then implies that ~ and ~ for all .
If #= and ~¸# 0¹ is a basis for , then we may write
8
=
#~ #
~
for s . Since the coefficients are real, we have
#b ~ ²# b ³
~
and so the coordinate matrices are equal:
´# b µ cpx²³8 ~ ´#µ 8
Exercises
=
1. Let be a vector space over . Prove that # ~ and ~ for all #=
-
and - . Describe the different 's in these equations. Prove that if
# ~ , then ~ or # ~ . Prove that # ~# implies that #~ or ~ .
