I was given the theorem
The solution set of a homogeneous system is a subspace of $\mathbb R^n$, called the solution space
and I was told to prove it in at least 2 different ways. I know one way to prove it is through the subspace test, but I'm having trouble thinking about another way to prove this theorem
Either way you would invoke subspace test, but it might be just deeply buried. Here is one way I can think of. Let's $A$ be a matrix $m\times n$ and equations $Ax=0$ have $k$ free variables. Each solution then is defined by the set of $k$ numbers. Thus, the solution is isomorphic to $\mathbb{R}^k$. We know that $\mathbb{R}^k$ is a subspace of $\mathbb{R}^n$ and thus the solution is a subspace in the space of all vectors. But I am basically deferring the subspace test (for $\mathbb{R}^k$ in $\mathbb{R}^n$).