The NullSpace(A) command computes a basis for the nullspace (kernel) of the linear transformation defined by Matrix A.  The result is a (possibly empty) set of Vectors.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$A≔⟨⟨1\,2\,3\,4⟩\left|⟨5\,6\,7\,8⟩\right|⟨9\,10\,11\,12⟩|⟨13\,14\,15\,16⟩⟩\:$

$\mathrm{kern}≔\mathrm{NullSpace}\left(A\right)$

$\mathrm{kern}≔\left\{\left[\begin{array}{c}2\\ \mathrm{-3}\\ 0\\ 1\end{array}\right]\,\left[\begin{array}{c}1\\ \mathrm{-2}\\ 1\\ 0\end{array}\right]\right\}$

$A·\mathrm{kern}\left[1\right]$

$\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$

$A·\mathrm{kern}\left[2\right]$

$\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$

$\mathrm{NullSpace}\left(\mathrm{Id}\left(3\right)\right)$

$\varnothing$