Page 144 - A Course in Linear Algebra with Applications

128 Chapter Five: Basis and Dimension

Example 5.2.1
Consider the matrix
3
2
( - 1 2 1 1 1 1 3 2 \ •
o o i o i
0 1 0 1 1 /

The reduced row echelon form of A is found to be
/ l 0 0 1 0\
0 1 0 1 1
o o i o i •
VO 0 0 0 0/
Hence the row vectors [1 0 0 1 0], [0 1 0 1 1], [0 0 1 0 1] form
a basis of the row space of A and the dimension of this space
is 3.

In general elementary row operations change the column
space of a matrix, and column operations change the row
space. However it is an important fact that such operations
do not change the dimension.
Theorem 5.2.3
For any matrix, elementary row operations do not change the
dimension of the column space and elementary column opera-
tions do not change the dimension of the row space.
Proof
Take the case of row operations first. Let A be a matrix
with n columns and suppose that B = EA where E is an
elementary matrix. We have to show that the column spaces
of A and B have the same dimension. Denote the columns
of A by Ai, A 2,. . ., A n. If some of these columns are linearly
dependent, then there are integers i± < i 2 < • • • < i r and
non-zero scalars c^, Cj 2 ,..., c ir such that

c^A^ +c i2A i2-\ h c ir A ir = 0.