[Math] Show or give counterexample: Every subspace of $\mathbb{R}^4$ is the nullspace of some matrix.

Question

Show or give counterexample: Every subspace of $\mathbb{R}^4$ is the nullspace of some matrix.

Would it be valid to just state that every subspace of $\mathbb{R}^n$ can be described as the null space of some matrix. Is there a counterexample?

I attempted showing the statement true in a concise manner. Does this work?

Let the basis $\{a_1$,$a_2$,$a_3$,$a_4\}$ be subspace S of $\mathbb{R}^4$.

Then let some matrix $A= \begin{bmatrix}
a_1 \\
a_2 \\
a_3 \\
a_4
\end{bmatrix}$

The row vectors of A are a basis for the null space of subspace S.

null(S)= A

Answer 1

Let $V$ be a vector space and $U$ a subspace of $V$. Then there is a subsace $W$ such that

$V= U \oplus W$. Define the linear mapping $P:V \to V$ as follows:

for $v \in V$ there are unique $u \in U$ and $w \in W$ such that $v=u+w$; put

$$Pv=w.$$

Then: $ker(P)=U$.

Can you get it from here ?