Homomorphismenräume/Hintereinanderschaltung/Multilinear/Aufgabe/Lösung

Es sei zunächst ${}\varphi \in \operatorname {Hom} _{K}{\left(U,V\right)}$ fixiert. Für ${}\psi _{1},\psi _{2}\in \operatorname {Hom} _{K}{\left(V,W\right)}$ und einen beliebigen Vektor ${}u\in U$ ist

${}{\begin{aligned}{\left({\left(\psi _{1}+\psi _{2}\right)}\circ \varphi \right)}(u)&={\left(\psi _{1}+\psi _{2}\right)}{\left(\varphi (u)\right)}\\&=\psi _{1}{\left(\varphi (u)\right)}+\psi _{2}{\left(\varphi (u)\right)}\\&={\left(\psi _{1}\circ \varphi \right)}(u)+{\left(\psi _{1}\circ \varphi \right)}(u)\end{aligned}}$

und für ${}s\in K$ ist

${}{\begin{aligned}{\left({\left(s\psi \right)}\circ \varphi \right)}(u)&={\left(s\psi \right)}{\left(\varphi (u)\right)}\\&=s{\left(\psi {\left(\varphi (u)\right)}\right)}\\&=s{\left(\psi \circ \varphi \right)}(u).\end{aligned}}$

Es sei nun ${}\psi \in \operatorname {Hom} _{K}{\left(V,W\right)}$ fixiert und ${}\varphi _{1},\varphi _{2}\in \operatorname {Hom} _{K}{\left(U,V\right)}$. Es ist

${}{\begin{aligned}{\left(\psi \circ {\left(\varphi _{1}+\varphi _{2}\right)}\right)}(u)&=\psi {\left({\left(\varphi _{1}+\varphi _{2}\right)}(u)\right)}\\&=\psi {\left(\varphi _{1}(u)+\varphi _{2}(u)\right)}\\&=\psi {\left(\varphi _{1}(u)\right)}+\psi {\left(\varphi _{2}(u)\right)}\\&={\left(\psi \circ \varphi _{1}\right)}(u)+{\left(\psi \circ \varphi _{2}\right)}(u)\\&={\left(\psi \circ \varphi _{1}+\psi \circ \varphi _{2}\right)}(u)\end{aligned}}$

und

${}{\begin{aligned}{\left(\psi \circ {\left(s\varphi \right)}\right)}(u)&=\psi {\left({\left(s\varphi \right)}(u)\right)}\\&=\psi {\left(s\varphi (u)\right)}\\&=s\psi {\left(\varphi (u)\right)}\\&=s{\left({\left(\psi \circ \varphi \right)}(u)\right)}\\&={\left(s{\left(\psi \circ \varphi \right)}\right)}(u).\end{aligned}}$

Zur gelösten Aufgabe