Proposition (Linearity properties for matrices of linear maps)

Let $V,W$ be finite dimensional vector spaces, with bases $\{v_1,...,v_n\}\subset V$ and $\{w_1,...,v_m\}\subset W$. If we have linear maps $S,T\in\mathscr{L}(V,W)$, then $$\mathcal{M}(S+T)=\mathcal{M}(S)+\mathcal{M}(T)$$ If $\lambda\in\mathbb{F}$, then $$\mathcal{M}(\lambda T)=\lambda \mathcal{M}(T)$$