[lin.alg]

Let $V$ be a $K$-vector space, then $A := \mathrm{span}\left\{ \text{linear maps}\ T : K \rightarrow V \right\} \cong V$ as vector spaces. (Abusing notation and treating $K$ as a 1D vector space over $K$ in the definition of $A$).

$A$ is a vector space. Easy to check that $\varphi : A \rightarrow V : T \mapsto Te$ where $e$ is the single basis vector in $K$ is a linear transformation. It is surjective: $\forall v \in V,\ T_v : K \rightarrow V,\ T_v(e) = v$ is a linear map. It is injective: if $Te = Se$ then $Ta = Tke = kTe = kSe = Sa\ \forall a \in K$, i.e. $S = T$.