(Solved): Find bases for the column space, the row space, and the null space of matrix A. You should verify ...

Find bases for the column space, the row space, and the null space of matrix A. You should verify that the Rank-Nullity Theorem holds. An equivalent echelon form of matrix \( \mathrm{A} \) is given to make your work easier. \[ A=\left[\begin{array}{ccc} 4 & 31 & 5 \\ 3 & 1 & 0 \\ -4 & -1 & 5 \\ 3 & 7 & -2 \\ -4 & 11 & 6 \end{array}\right] \sim\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \] Basis for the column space of \( A \) is Basis for the row space of \( A \) is Note that since the only solution to \( A \mathbf{x}=\mathbf{0} \) is the zero vector, there is no basis for the null space of \( A \).