Linear maps

A linear map $T$ is a function (= map, mapping) between two vector spaces $V$ and $W$ over a field $\mathbb{F}$: $$T:V\to W,$$ such that $\mathrm{\forall }{\mathbf{v}}_1,{\mathbf{v}}_2\in V;\mathrm{\forall }a\in \mathbb{F}:$ $$\begin{array}{rlrl}& T({\mathbf{v}}_1+{\mathbf{v}}_2)=T({\mathbf{v}}_1)+T({\mathbf{v}}_2)& & T\text{ ‘preserves’ vector addition}\\ & T(a{\mathbf{v}}_1)=aT({\mathbf{v}}_1)& & T\text{ ‘preserves’ scalar multiplication}\end{array}$$

We will call the set of all linear maps from $V$ to $W$ $\mathcal{L}(V,W)$. If we define the sum of two linear maps $S$ and $T$ and the scalar product of an element $a\in \mathbb{F}$ with a linear map $T$ such that $\mathrm{\forall }\mathbf{v}\in V:$ $$\begin{array}{rl}& (S+T)(\mathbf{v})=S(\mathbf{v})+T(\mathbf{v})\\ & (aS)(\mathbf{v})=a(S(\mathbf{v})),\end{array}$$ respectively, we can show that $\mathcal{L}(V,W)$ forms a vector space over the field $\mathbb{F}$.

A linear operator is a linear map $T$ from a vector space $V$ to itself: $$T:V\to V.$$ Obviously, $\mathcal{L}(V)=\mathcal{L}(V,V)$ also forms a vector space over $\mathbb{F}$.