Page 67 - Advanced Linear Algebra
P. 67
Vector Spaces 51
#b Ä b # ~
where # 7 r 8 r 9 and £ for all . There must be vectors in this
#
expression from both and , since 7 r 8 and r 8 9 are linearly independent.
9
7
Isolating the terms involving the vectors from on one side of the equality
7
shows that there is a nonzero vector in %º » q º r » . But then %: q ;
7
9
8
7
8
and so %º » q º » , which implies that % ~ , a contradiction. Hence,
7 8 r 9 r is linearly independent and a basis for : b ; .
Now,
dim²:³ b dim²;³ ~ ( r (7 b ( 8 r ( 8 9
~ 7 b( ( b (( 8 b (( 8 ( (
9
~ 7 b (( b ( ( 8 b ( ( 9 dim : ² q ; ³
~ dim²:b ;³ b dim²:q ;³
as desired.
It is worth emphasizing that while the equation
dim²:³ b dim²;³ ~ dim²: b ;³ b dim²: q ;³
holds for all vector spaces, we cannot write
dim²: b ;³ ~ dim²:³ b dim²;³ c dim²: q ;³
unless :b ; is finite-dimensional.
Ordered Bases and Coordinate Matrices
It will be convenient to consider bases that have an order imposed on their
members.
Definition Let be a vector space of dimension . An ordered basis for is
=
=
an ordered -tuple ²# Á ÃÁ# ³ of vectors for which the set ¸# ÁÃÁ# ¹ is a
basis for .
=
If 8 ~²# Á Ã Á # ³ is an ordered basis for , then for each # = there is a
=
unique ordered -tuple ² Á Ã Á ³ of scalars for which
# ~ # bÄb #
Accordingly, we can define the coordinate map 8 ¢= ¦ - by
v y
8 ²#³ ~ ´#µ ~ Å (1.3 )
8
w z
