Linear form/Vector/Composition/Remark

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

${\displaystyle f\colon V\longrightarrow K}$

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

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

is linear. It is just the composition

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

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