$\underline{Def}$
Let $V,W$ be vector space over $\mathbb{F}$ and let $a \in \mathbb{F}$.
Let $T,U \colon V \rightarrow W$ be any maps.
We define two maps $T+U,aT \colon V \rightarrow W$ by
$$(T+U)(v)=T(v)+U(v)$$
$$(aT)(v)=aT(v)\,for\, v \in V$$
$\underline{Rmk}$
Let $\mathcal{F}(V,W)$ be the set of all maps from $V$ into $W$. With the operations of additaion & scalar multiplication above, $\mathcal{F}(V,W)$ is a vector space over $\mathbb{F}$. Its zero vector is the zero map $T_{0} \colon V \rightarrow W$.
$\underline{Def}$
Let $V,W$ be vector spaces over $\mathbb{F}$. Let $\mathcal{L}(V,W)$ denote the set of all linear maps from $V$ to $W$.
(Thus, $\mathcal{L}(V,W) \subset \mathcal{F}(V,W)$)
$\underline{Thm}$
$\mathcal{L}(V,W)$ is a subspace of $\mathcal{F}(V,W)$, that is, $\mathcal{L}(V,W)$ is a vector space over $\mathbb{F}$ itself.
$\underline{Rmk}$
Let $V,W$ be finite-dimensional vector spaces over $\mathbb{F}$ with ordered basis $\beta,\gamma$, respectively. Let $n=\vert n \vert,m=\vert \gamma \vert$.
$$\mathcal{L}(V,W)\;\;\;\;\;\;\; M_{m \times n}(\mathbb{F})$$
$$T \longleftrightarrow [T]_{\beta}^{\gamma}$$
$$ addition \longleftrightarrow addition$$
$$ scalar\, multiplication \longleftrightarrow scalar\, multiplication$$