Homomorphism space/Linear subspaces/Fact

{{ Mathematical text/Fact |Text= {{ Factstructure|typ= |Situation=Let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be vector spaces over a field ${\displaystyle {}K}$. |Condition=

|Segue= Then the following subsets are linear subspaces of ${\displaystyle {}\operatorname {Hom} _{K}{\left(V,W\right)}}$. |Conclusion= {{ Enumeration4 |For a linear subspace ${\displaystyle {}U\subseteq V}$, {{ Relationchain/display |S || {{Setcond| \varphi \in \operatorname{Hom}_{ K } { \left( V , W \right) } | \varphi {{|}}_U = 0 }} || || || |pm= }} is a linear subspace of ${\displaystyle {}\operatorname {Hom} _{K}{\left(V,W\right)}}$. If ${\displaystyle {}V}$ and ${\displaystyle {}W}$ are finite-dimensional, then

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

|For a linear subspace ${\displaystyle {}U\subseteq W}$,

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

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

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

|For linear subspaces ${\displaystyle {}V_{1}\subseteq V}$ and ${\displaystyle {}W_{1}\subseteq W}$,

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

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

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

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

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

is a linear subspace of ${\displaystyle {}\operatorname {Hom} _{K}{\left(V,W\right)}}$. }} |Extra= }} |Textform=Fact |Category= |Request=Linear subspaces in the space of homomorphisms }}