Final answer:
To find a basis for the row space and the rank of a matrix, perform row reduction to obtain the row-echelon form. The row space is the span of the non-zero rows in the row-echelon form, and the rank is the number of linearly independent rows.
Step-by-step explanation:
To find a basis for the row space and the rank of a matrix, we need to perform row reduction to obtain the row-echelon form or the reduced row-echelon form of the matrix. Let's perform row reduction on the given matrix:
-3 -12 12
5 2 8
-8 -9 -3
-12 12 4
By performing row reduction, we obtain the following row-echelon form:
1 4 -4
0 3 3
0 0 0
0 0 0
The row space is the span of the non-zero rows in the row-echelon form. In this case, the row space is spanned by the vectors [1, 4, -4] and [0, 3, 3]. Therefore, a basis for the row space is { [1, 4, -4], [0, 3, 3] }.
The rank of a matrix is the number of linearly independent rows or columns. In this case, the rank is equal to the number of non-zero rows in the row-echelon form, which is 2. Therefore, the rank of the given matrix is 2.