Page 68 - Advanced Linear Algebra

52 Advanced Linear Algebra

is known as the coordinate matrix of with
#
where the column matrix ´#µ 8
8
respect to the ordered basis . Clearly, knowing ´#µ 8 is equivalent to knowing #
(assuming knowledge of 8 ).
is bijective and
Furthermore, it is easy to see that the coordinate map 8
preserves the vector space operations, that is,
8 8 ² # b Äb ²# ³bÄb # ³ ~ 8 ²# ³

or equivalently
8
8

´ # b Äb # µ ~ ´# µ bÄb ´# µ 8

Functions from one vector space to another that preserve the vector space
operations are called linear transformations and form the objects of study in the
next chapter.
The Row and Column Spaces of a Matrix
Let be an d matrix over . The rows of span a subspace of - known
(
-
(
as the row space of and the columns of span a subspace of - known as
(
(
the column space of . The dimensions of these spaces are called the row rank
(
and column rank , respectively. We denote the row space and row rank by
rs²(³ and rrk²(³ and the column space and column rank by cs²(³ and crk²(³.
It is a remarkable and useful fact that the row rank of a matrix is always equal to
its column rank, despite the fact that if £ , the row space and column space
are not even in the same vector space!
Our proof of this fact hinges on the following simple observation about
matrices.
Lemma 1.15 Let be an d matrix. Then elementary column operations do
(
not affect the row rank of . Similarly, elementary row operations do not affect
(
the column rank of .
(
Proof. The second statement follows from the first by taking transposes. As to
the first, the row space of is
(
rs²(³ ~ º (Á Ã Á (»

where are the standard basis vectors in - . Performing an elementary
column operation on ( is equivalent to multiplying ( on the right by an
,
elementary matrix . Hence the row space of , ( is
rs²(,³ ~ º (,Á Ã Á (,»

and since is invertible,
,