Providing a counterexample to show the Nul(A) is a subspace of R^n

Question

So in my linear algebra class, we are supposed to give a counterexample if some true/false question is false. The question given is,

The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of $R^m$.

Now, I know that this is false because the $Nul(A)$ is actually a subspace of $R^n$.

However, this would not be an acceptable answer. It is false, so i need some counter-example, and I'm not quite sure how to come up with a counter-example for this problem.

Answer 1

Consider the case when $n=1$ and $m=8$. The set of solutions is a member of $R^1$ and will not have 8 dimensions.

Answer 2

Let $\mathbf{A}=\begin{bmatrix}1 & 1\end{bmatrix}$.

Given $\mathbf{A}\mathbf{x}=\mathbf{0}$, any solution, $\mathbf{x}$, must be an element in $\mathbb{R}^2$ by the definition of matrix-vector multiplication. But $\mathbf{A}\mathbf{x}=\mathbf{0}$ corresponds to a system with only 1 equation. Thus the given statement must be false.