Homomorphism space/Linear subspaces/Fact

Let ${}V$ and ${}W$ be vector spaces over a field ${}K$. Then the following subsets are linear subspaces of ${}\operatorname {Hom} _{K}{\left(V,W\right)}$.

1. For a linear subspace ${}U\subseteq V$,

   ${}S={\left\{\varphi \in \operatorname {Hom} _{K}{\left(V,W\right)}\mid \varphi {|}_{U}=0\right\}}\,$

   is a linear subspace of ${}\operatorname {Hom} _{K}{\left(V,W\right)}$. If ${}V$ and ${}W$ are finite-dimensional, then

   ${}\dim _{K}{\left(S\right)}={\left(\dim _{K}{\left(V\right)}-\dim _{K}{\left(U\right)}\right)}\dim _{K}{\left(W\right)}\,.$

2. For a linear subspace ${}U\subseteq W$,

   ${}T={\left\{\varphi \in \operatorname {Hom} _{K}{\left(V,W\right)}\mid \operatorname {Im} \varphi \subseteq U\right\}}\,$

   is a linear subspace of ${}\operatorname {Hom} _{K}{\left(V,W\right)}$, which is isomorphic to ${}\operatorname {Hom} _{K}{\left(V,U\right)}$. If ${}V$ and ${}W$ are finite-dimensional, then

   ${}\dim _{K}{\left(T\right)}=\dim _{K}{\left(V\right)}\cdot \dim _{K}{\left(U\right)}\,.$

3. For linear subspaces ${}V_{1}\subseteq V$ and ${}W_{1}\subseteq W$,

   ${}H={\left\{\varphi \in \operatorname {Hom} _{K}{\left(V,W\right)}\mid \varphi (V_{1})\subseteq W_{1}\right\}}\,$

   is a linear subspace of ${}\operatorname {Hom} _{K}{\left(V,W\right)}$. If ${}V$ and ${}W$ finite-dimensional, then

   $\dim _{K}{\left(H\right)}=\dim _{K}{\left(V_{1}\right)}\dim _{K}{\left(W_{1}\right)}+{\left(\dim _{K}{\left(V\right)}-\dim _{K}{\left(V_{1}\right)}\right)}\dim _{K}{\left(W\right)}\,.$

4. For linear subspaces ${}V_{1},\ldots ,V_{n}\subseteq V$ and ${}W_{1},\ldots ,W_{n}\subseteq W$,

   ${}L={\left\{\varphi \in \operatorname {Hom} _{K}{\left(V,W\right)}\mid \varphi (V_{1})\subseteq W_{1}{\text{ and }}\varphi (V_{2})\subseteq W_{2}{\text{ and }}\ldots {\text{ and }}\varphi (V_{n})\subseteq W_{n}\right\}}\,$

   is a linear subspace of ${}\operatorname {Hom} _{K}{\left(V,W\right)}$.

Proof, Write another proof