Page 231 - Advanced engineering mathematics

7.4 Row and Column Spaces 211

To see this, recall that the nonzero rows of R form a basis for the row space of this matrix, hence
their number is the dimension of this row space.
Now suppose we perform an elementary row operation on A to form B. How does this
change the row space of A? The answer is that it does not change it at all. This fact will be
important in solving systems of linear equations

THEOREM 7.10

Let B be formed from an n × m matrix A by a sequence of elementary row operations. Then A
and B have the same row space, hence also
rank(A) = rank(B).
Proof It is enough to prove the theorem for the case that B is formed from A by one elementary
row operation. Let the row vectors of A be A 1 ,··· ,A n . The row space of A is the subspace of
m
R consisting of all linear combinations
α 1 A 1 + α 2 A 2 + ··· + α n A n .
If the elementary row operation is an interchange of rows, then the rows of A and B are the
m
same (appearing in a different order) and hence span the same subspace of R .
Suppose a type II elementary row operation is performed, multiplying row r of A by the
nonzero number c. Now the row space of B consists of all vectors

α 1 A 1 + ··· + cα r A r + ··· + α n A n .
Since the α j ’s are arbitrary, this is again a linear combination of the rows of A, hence the row
spaces of A and B are the same.
Finally, consider the case that a type III operation is performed, adding c times row i to row
j to form B. Now the row vectors of B are
A 1 ,··· ,A j−1 ,cA i + A j ,A j+1 ,··· ,A n .
Any linear combination of these rows is again a linear combination of the rows of A, hence in
this case the row spaces of A and B are also the same.
Finally, because the row spaces are the same, their dimension is the same and the matrices
have the same rank.

If we defined elementary column operations analogous to the elementary row operations, we
would find that these leave the column space of a matrix unchanged.
Theorem 7.10 has several important consequences.

COROLLARY 7.1

For any real matrix A, A and A R have the same row space. Thus,
rank(A) = number of nonzero rows of A R .
This follows from the fact A R is formed from A by a sequence of elementary row
operations, so
rank(A) = rank(A R )
= number of nonzero rows of A R .

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October 14, 2010 14:23 THM/NEIL Page-211 27410_07_ch07_p187-246