Let S be the subspace of R4 given by the solution set of the equations
-x3 + 4 x4 = -x2 - x3 + x4 and -x1 - x3 = -x1 - 3 x3 = x2
An example of a matrix for which S is the nullspace is....
i tried doing 1 1 0 0
0 1 0 3
$\displaystyle -x_3 + 4 x_4 = -x_2 - x_3 + x_4 \Rightarrow x_2 = -3 x_4$ .... (1)
$\displaystyle -x_1 - x_3 = -x_1 - 3 x_3 \Rightarrow x_3 = 0$ .... (2)
$\displaystyle -x_1 - x_3 = x_2 \Rightarrow x_1 = -x_3 - x_2$ .... (3)
Substitute from (1) and (2): $\displaystyle x_1 = 3 x_4$.
So elements of S have the form $\displaystyle x_4 (3, -3, 0, 1)$.
So the example you propose works.