\[T\]
dimension of the image space is the number of non zero rows in the row reduced form of the matrx and the dimension of the kernel is the number of zero rows in the row reduced formExample:
\[\mathbf{M} = \left( \begin{array}{ccc} 1 & 2 & 1 \\ 1 & -1 & 0 \\ -1 & -2 & -1 \\ 2 & 1 & 1 \end{array} \right) \]
\[ \left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 3 & -1 \end{array} \right) \begin{array}{c} \\ R2-R1 \\ R3+R1 \\R4-2*R1 \end{array}\]
\[ \left( \begin{array}{ccc} 1 & 2 & 1 \\ 0 & -3 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \begin{array}{c} \\ \\ \\R4-R2 \end{array}\]
There are two non zero rows so the dimension of the image space is 2 and there are two zero rows so the dimension of the kernel is 2.