Page 229 - Advanced engineering mathematics

7.4 Row and Column Spaces 209

EXAMPLE 7.16
Let
⎛ ⎞
5 −1 5
−1 1 3
⎜ ⎟
⎜ ⎟
A = ⎜ 1 1 7 ⎟ .
⎜ ⎟
⎝ 2 0 4 ⎠
1 −3 −6
3
The row space is the subspace of R spanned by the five rows of A. This subspace consists of all
linear combinations
α(5,1,−5) + β(−1,1,3) + γ(1,1,7)

+ δ(2,0,4) + (1,−3,6)
of the row vectors. The last three row vectors are linearly independent (none is a linear
combination of the other two). The first two are linear combinations of the last three:

(5,−1,5) =−(1,1,7) + 3(2,0,4)
and

(−1,1,3) = (1,1,7) − (2,0,4),

The first three row vectors therefore form a basis for the row space. This row space has dimension
3
3 and is all of R . The row rank of A is 3.
The column space of A is the subspace of R consisting of all linear combinations of the
5
column vectors, which we continue to write as columns:
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
5 −1 5
−1 1 3
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
α ⎜ 1 ⎟ + β ⎜ 1 ⎟ + γ ⎜ 7 ⎟ .
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ 2 ⎠ ⎝ 0 ⎠ ⎝ 4 ⎠
1 −6 −6
5
These three column vectors are linearly independent in R and span a three-dimensional subspace
5
of R . The column rank of A is 3.
In this example,
row rank of A = column rank of A = 3.

We claim that this is not a coincidence.

THEOREM 7.9 Equality of Row and Column Rank
For any matrix, the row rank equals the column rank.

Proof Although this is true in general, we will prove it when each a ij is a real number, enabling
us to exploit the row and column spaces of A. Suppose A is n × m:

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:23 THM/NEIL Page-209 27410_07_ch07_p187-246