generally, steps of calculating kernel & image are getting rref(matrix) at first

Image:

$\displaystyle

rref\left[ \begin{array}{cccc} -1 & 0 & 0 & 1 \\1 & -1 & 0 & 0 \\-1 & 0 & 1 & 0 \\0 & 1 & 0 & -1 \\0 & 0 & -1 & 1 \\0 & -1 & 1 & 0 \end{array} \right]=\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 \end{array} \right]$

find columns including leading 1's in rref(matrix), these columns are linearly independent.

im(A)=span(column1, \cdots , column4)