Linear Algebra/K/Homomorphism space/Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be ${\displaystyle {}K}$-vector spaces. Then

${\displaystyle {}\operatorname {Hom} _{K}{\left(V,W\right)}={\left\{f:V\rightarrow W\mid f{\text{ linear mapping}}\right\}}\,}$

is called the space of homomorphisms. It is endowed with the addition defined by

${\displaystyle {}(f+g)(v):=f(v)+g(v)\,,}$

and the scalar multiplication defined by

${\displaystyle {}(\lambda f)(v):=\lambda \cdot f(v)\,.}$