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210 CHAPTER 7 Matrices and Linear Systems

⎛ ⎞
a 11 a 12 ··· a 1r a 1,r+1 ··· a 1m
⎜ a 21 ··· ··· ⎟
a 22 a 2r a 2,r+1 a 2m
⎜ . . . . . . ⎟
⎜
⎜ . . . . . . . . . . . . . . ⎟
.
⎟
⎜ ⎟
A = ⎜ a r1 a r2 ··· a rr a r,r+1 ··· a rm ⎟ .
⎜ ⎟
a r+1,1 a r+1,2 ··· a r+1,r a r+1,r+1 ··· a r+1,m
⎜ ⎟
⎜ . . . . . . . ⎟
⎜
⎝ . . . . . . . . . . . . . . ⎟
⎠
a n1 a n2 ··· a nr a n,r+1 ··· a nm
Denote the row vectors R 1 ,R 2 ,··· ,R n ,so
m
R i = (a i1 ,a i2 ,··· ,a im ) in R .
Suppose the row rank of A is r. As a notational convenience, suppose the first r rows
are linearly independent. Then each of R r+1 ,··· ,R n is a linear combination of R 1 ,··· ,R r .
Write
R r+1 = β r+1,1 R 1 + ··· + β r+1,r R r
R r+2 = β r+2,1 R 1 + ··· + β r+2,r R r
.
.
.
R n = β n,1 R 1 + ··· + β n,r R r .
Now observe that column j of A can be written
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
a 1 j 1 0 0
⎜ a 2 j ⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟
. . . .
⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
0 0 1 .
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
a rj
⎜ ⎟ = a 1 j ⎜ ⎟ + a 2 j ⎜ ⎟ + ··· + a rj ⎜ ⎟
a r+1, j β r+1,1 β r+1,2 β r+1,r
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟
.
.
.
.
⎝ . ⎠ ⎝ . ⎠ ⎝ . ⎠ ⎝ . ⎠
a nj β n1 β n,2 β n,r
Thus, each column of A is a linear combination of the rn-vectors on the right side of the last
equation. These r vectors therefore span the column space of A, so the dimension of this column
space is at most r (equal to r if these columns are linearly independent, less than r if they are
not). This proves that
dimension of the column space of A ≤ dimension of the row space.
By repeating this argument, using columns instead of rows, we find that the dimension of
the row space is less than or equal to the dimension of the column space. This proves the
theorem.
Now define the rank of A as the row rank of the matrix, which is the same as the column
rank. Denote this number as rank(A). The matrix of Example 7.16 has rank 3.

Given an arbitrary real matrix A, it may not be obvious what the rank of a A is. However,
if R is a reduced matrix, then
rank(R) = number of nonzero rows of R.

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October 14, 2010 14:23 THM/NEIL Page-210 27410_07_ch07_p187-246