Addition and Scalar Multiplication on Sets of Linear Maps

Addition and scalar multiplication on $\mathcal{L}(V,W)$

Suppose $S,T\in \mathcal{L}(V,W)$ and $\lambda \in \mathbb{F}$. The sum $S+T$ and product $\lambda T$ are the linear maps from $V$ to $W$ defined by

$$(S+T)v=Sv+Tv\text{and}(\lambda T)(v)=\lambda (Tv)$$

for all $v\in V$.

We can verify $S+T$ and $\lambda T$ as defined above are linear maps. In other words, if $S,T\in \mathcal{L}(V,W)$ and $\lambda \in \mathbb{F}$, then $S+T\in \mathcal{L}(V,W)$ and $\lambda T\in \mathcal{L}(V,W)$.

The above addition and scalar multiplication leads us to this result:

$\mathcal{L}(V,W)$ is a vector space

With the operations of addition and scalar multiplication defined above, $\mathcal{L}(V,W)$ is a vector space.