Vector spaces of linear transformations

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The collection of all linear transformations between given vector spaces itself forms a vector space.

Suppose $f$ and $g$ are linear transformations from $V$ to $W$ . Then the function $f + g : V \to W$ defined by the rule which sends $v \in V$ to $f(v) + g(v)$ is also a linear transformation.

Similarly, if $f : V \to W$ is linear and $\alpha$ is a scalar, then the function $\lambda f : V \to W$ defined by the rule $(\lambda f)(v) = \lambda f(v)$ is also a linear transformation.

Suppose $V$ and $W$ are vector spaces. The collection $Lin(V,W)$ of linear transformations from $V$ to $W$ forms a vector space under the above operations.

In fact, using bases we can say something more. If $S$ is a basis for an $n$ -dimensional vector space $V$ , and $T$ a basis for an $m$ -dimensional vector space $W$ , then as we have seen above this data can be used to associate to the linear transformation $L:V\to W$ an $m\times n$ matrix $L\mapsto \phi _{S,T}(L) := {}_T L_S\in \mathbb R^{m\times n}$ which is essentially the coordinate representation of $L$ with respect to the pair of bases $S,T$ . It is easily seen that under the above association

$\begin{align*} &\phi_{S,T}(L_1 + L_2) = {}_T (L_1)_S + {}_T (L_2)_S\\ & \phi_{S,T}(\alpha L) = \alpha ({}_T L_S) \end{align*}$

In other words,

Given a basis $S$ for an $n$ -dimensional vector space $V$ and a basis $T$ for an $m$ -dimensional vector space $W$ , the map $\phi _{S,T}: Lin(V,W)\to \mathbb R^{m\times n}$ is an isomorphism between the vector space of linear transformations from $V$ to $W$ and the vector space of $m\times n$ matrices with entries in $\mathbb R$ .