Linear form/Vector/Composition/Remark

Let ${}K$ be a field, and let ${}V$ and ${}W$ denote vector spaces over ${}K$. For a linear form

$f\colon V\longrightarrow K$

and a vector ${}w\in W$, the mapping

$fw\colon V\longrightarrow W,v\longmapsto f(v)w,$

is linear. It is just the composition

$V{\stackrel {f}{\longrightarrow }}K{\stackrel {\iota _{w}}{\longrightarrow }}W,$

where ${}\iota _{w}$ denotes the mapping ${}s\rightarrow sw$.