Let A 2 4 8 4 5 13 5 5 5 4 8 0 5 11 1 Find a basis for the

Let A = [-2 4 -8 -4 5 -13 5 5 5 -4 -8 0 5 11 -1]. Find a basis for the row space of A.

Solution

First, we must convert the matrix to reduced row echelon form:

Divide row1 by -2

Add (-4 * row1) to row2

Add (8 * row1) to row3

Divide row2 by -3

Add (-3 * row2) to row3

Add (-2 * row2) to row1

Because we have only performed linear operations on rows, the non-zero rows in the reduced row echelon form of the matrix comprise a Basis for the Row Space of the matrix.
(Note that this is not true of the Column Space; the Column Space certainly changes as you perform row operations.)
The rows highlighted below in BOLD comprise a Basis for the Row Space of our matrix:

1	2	-5/2	2	-5/2
4	5	5	-8	11
-8	-13	5	0	-1