Final answer:
The rank of the matrix is 2
To find a basis for the column space and the rank of a matrix, we need to find the linearly independent columns of the matrix. The rank of a matrix is the number of linearly independent columns it has.
Step-by-step explanation:
To find a basis for the column space of a matrix, we need to find the linearly independent columns of the matrix.
The rank of a matrix is the number of linearly independent columns it has.
In this case, we have the matrix:
2 4 -3 -6
-2 -4 1 -2
2 4 -2 -2
7 14 -6 -3
First, we row reduce the matrix to find its reduced row echelon form:
1 2 0 1
0 0 1 -1
0 0 0 0
0 0 0 0
The pivot columns (columns with leading 1's) are the linearly independent columns of the matrix. In this case, the first and third columns are the pivot columns, so a basis for the column space is:
[2, -2, 2, 7] and [-3, 1, -2, -6]
Therefore, the rank of the matrix is 2.