Linear Algebra/K/Homomorphism space/Definition

Space of homomorphisms

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

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

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

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

and the scalar multiplication defined by

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