158 CHAPTER 8. RANK OF A MATRIX

The row reduced echelon form is1 0 0 0 13

20 1 0 2 − 5

20 0 1 −1 1

20 0 0 0 0

 .

and so the rank is 3, the row space is the span of the vectors,(0 0 1 −1 1

2

),(

0 1 0 2 − 52

),(

1 0 0 0 132

),

and the column space is the span of the first three columns in the original matrix,

span



1111

 ,

2323

 ,

1612

 .

Example 8.3.8 Find the rank of the following matrix and describe the column and rowspaces efficiently.  1 2 3 0 1

2 1 3 2 4−1 2 1 3 1

 .

The row reduced echelon form is 1 0 1 0 2117

0 1 1 0 − 217

0 0 0 1 1417

 .

It follows the rank is three and the column space is the span of the first, second and fourthcolumns of the original matrix.

span

 1

2−1

 ,

 212

 ,

 023



while the row space is the span of the vectors(0 0 0 1 14

17

),(

0 1 1 0 − 217

),(

1 0 1 0 2117

).

Procedure 8.3.9 To find the rank of a matrix, obtain the row reduced echelon form forthe matrix. Then count the number of nonzero rows or equivalently the number of pivotcolumns. This is the rank. The row space is the span of the nonzero rows in the rowreduced echelon form and the column space is the span of the pivot columns of the originalmatrix.