Find a basis for the column space and the rank of A 1 2 1 3

Find a basis for the column space and the rank of A = [1 2 -1 3 -2 -4 2 -6 1 -1 2 0 3 2 1 5]

Solution

Please apply the below Operations to find the basis of Column Space Matrix

Add (2 * row1) to row2

Add (-1 * row1) to row3

Add (-3 * row1) to row4

Swapping row3 with row2

Add (-4/3 * row2) to row4

First, we must reduce the matrix so we can calculate the pivots of the matrix (note that we are reducing to row echelon form, not reduced row echelon form):

The matrix has 2 pivots (hilighted above in yellow)
Because we have found pivots in columns 0 and 1. We know that these columns in the original matrix define the Column Space of the matrix.
Therefore, the Column Space is given by the following equation:

Since , the number of leading entries in the basis of column space matrix is 2 . Hence , the rank of the given matrix is also 2 .

1	2	-1	3
0	0	0	0
1	-1	2	0
3	2	1	5