Page 384 - Advanced Linear Algebra
P. 384
368 Advanced Linear Algebra
~ " n # 5 ~ ² k !³²" Á # ³~ ²" Á # ³
4
The key point is that this holds for any bilinear function ¢ < d = ¦ > . In
particular, let < i and = i and define by
²"Á #³ ~ ²"³ ²#³
which is easily seen to be bilinear. Then the previous display becomes
²" ³ ²# ³ ~
If, for example, the vectors are linearly independent, we can take to be a
"
dual vector " i to get
i
~ " ²" ³ ²# ³~ ²# ³
and since this holds for all linear functionals = i , it follows that # ~ . We
have proved the following useful result.
Theorem 14.5 If "Á Ã Á " are linearly independent vectors in < and
are arbitrary vectors in = , then
#Á Ã Á #
" n #~ ¬ #~ for all
In particular, " n #~ if and only if "~ or #~ .
Coordinate Matrices and Rank
If 8 is a basis for and ~ ¸# 1¹ is a basis for , then
<
=
9 ~¸" 0¹
any vector ' < n = has a unique expression as a sum
'~ ²" n # ³
Á
0 1
are nonzero. In fact, for a
where only a finite number of the coefficients Á
fixed ' < n = , we may reindex the bases so that
'~ ²" n # ³
Á
~ ~
where none of the rows or columns of the matrix 9~ ² ³ consists only of 's.
Á
The matrix 9~ ² ³ is called a coordinate matrix of with respect to the
'
Á
bases and .
9
8
Note that a coordinate matrix is determined only up to the order of its rows
9
and columns. We could remove this ambiguity by considering ordered bases,
