Questions: Find a basis for the row space and the rank of the matrix. [ [-3 1 2 10 6 5 -7 5 8] ]

Solution Steps

To find a basis for the row space of a matrix, we need to perform row reduction to bring the matrix to its row echelon form (REF) or reduced row echelon form (RREF). The non-zero rows of the resulting matrix will form a basis for the row space. The rank of the matrix is the number of non-zero rows in its REF or RREF.

We start with the matrix

\[ A = \begin{bmatrix} -3 & 1 & 2 \\ 10 & 6 & 5 \\ -7 & 5 & 8 \end{bmatrix} \]

and perform row reduction to obtain its reduced row echelon form (RREF).

After row reduction, we find the non-zero rows of the RREF matrix to form a basis for the row space. The non-zero rows are:

\[ \begin{bmatrix} 12.5698 & 1.7502 & -0.9547 \\ 0 & -7.6770 & -9.5963 \end{bmatrix} \]

The rank of the matrix is determined by the number of non-zero rows in the RREF. In this case, there are 2 non-zero rows, so the rank is

\[ \text{rank}(A) = 2 \]

Final Answer