Find a basis of the column space of the matrix A 2 14 0 3 1

Find a basis of the column space of the matrix A = [-2 1-4 0 3 1 3 -1 4 3 2-2].

Solution

We will reduce A to its RREF as under:

Multiply the 1st row by -1/2

Add -3 times the 1st row to the 2nd row

Add -4 times the 1st row to the 3rd row

Multiply the 2nd row by 2/5

Add -5 times the 2nd row to the 3rd row

Add 1/2 times the 2nd row to the 1st row

Then the RREF of A is

1

0

7/5

-1/5

0

1

-6/5

-2/5

0

0

0

0

The a basis for Col(A) is { (-2,3,4)^T, (1,1,3)^T} as the 3^rd and 4^th columns of A are linear combinations of the first two columns.

1	0	7/5	-1/5
0	1	-6/5	-2/5
0	0	0	0