Consider a possible system of m linear equations with n variables, on $\mathbb{K}$, in the form $AX = B$ ($A$,$B$ and $X$ are matrices). Show that the set C of the solutions is only a subspace of $\mathbb{K}^n$ if $B=0$, (in other words, if AX=B is an homogenous system).
In the book $\mathbb{K}$ is used for $\mathbb{R}$ and $\mathbb{C}$ interchangeably.
I think this is proven because of that condition that for a subspace to be a subspace of vectorial space it needs to contain its null vector. The trick here is probably to prove that in a homogenous system, (0,0,0,...) (whatever the degree of the subspace is) is necessarily a solution but I don't know how to do that. Help?
1.) The solutions of $AX =0$ are a subspace:
$0$ is a solution, since $A0=0$. If $X_1$ and $X_2$ are solutions, then $A(\lambda X_1 + X_2) = \lambda AX_1 + AX_2 = \lambda 0 + 0 = 0$. Thus, $\lambda X_1 + X_2$ is also a solution.
2.) The solutions of $AX = B$ are not a subspace for $B\neq 0$:
You already said it. The reason is, that $0$ is not a solution of $AX = B$, but must be contained in a subspace.
In general: If $AX = B$, then $\{Y | AY = B\} = X + \{Y|AY = 0\}$.