Final answer:
To find the basis for the row space and the rank of a matrix, perform row operations on the matrix and identify the nonzero rows.
Step-by-step explanation:
To find a basis for the row space and the rank of a matrix, we need to perform row operations on the matrix and identify the nonzero rows.
Given the matrix:
8 5 -7
2 1 -3
5 6 10
We can row reduce the matrix to:
1 0 -4
0 1 2
0 0 0
The nonzero rows in the row reduced form are the basis for the row space. Therefore, the basis for the row space is:
{1 0 -4, 0 1 2}
We can see that there are two nonzero rows in the row reduced form. Hence, the rank of the matrix is 2.