Computing the basis of a subspace of matrices with the given nullspace. Compute the dimension and find a basis for the subspace of matrices $H ⊂ M_{3×5} (\mathbb{R})$
such that $[0,2,−3,0,1]^T$ is in the nullspace of A ∈ H.
I have said that, if we want $[0,2,−3,0,1]^T$ to be in the nullspace of A, then we must be able to write,
$$c_1\vec{v_1}+...+c_k\vec{v_k} = [0,2,−3,0,1]^T $$
where $S = \{\vec{v_1},...,\vec{v_k}\}$ is a basis for the nullspace of A.
so we require
$$V\mathbb{\vec{c}} =  [0,2,−3,0,1]^T $$ to be consistent. where $V$ is a matrix whose columns are the elements of S.
How can I move on from here?
I feel as though this is wrong in the first place. Could someone give me push please.

$$\{\left( \begin{array}{ccc}
0 & -1 & 1 & 0 & 5\\
1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\end{array} \right),\left( \begin{array}{ccc}
0 & 1 & 1 & 0 & 1\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0\end{array} \right),\left( \begin{array}{ccc}
0 & 0 & 0 & 0 & 0\\
0 & 1 & 1 & 0 & 1\\
1 & 0 & 0 & 0 & 0\end{array} \right)
\left( \begin{array}{ccc}
0 & 0 & 0 & 0 & 0\\
0 & -1 & 1 & 0 & 5\\
0 & 0 & 0 & 1 & 0\end{array} \right)\left( \begin{array}{ccc}
1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 1 & 1 & 0 & 1\end{array} \right),\left( \begin{array}{ccc}
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & -1 & 1 &0 & 5\end{array} \right)\}$$
so $Dim(H) =6$
