Vector space/Linear subspace/Projection/Definition

Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and ${\displaystyle {}U\subseteq V}$ a linear subspace. A linear mapping

${\displaystyle \varphi \colon V\longrightarrow V}$

is called a projection of ${\displaystyle {}V}$ onto ${\displaystyle {}U}$, if ${\displaystyle {}U=\operatorname {Im} \varphi }$ and ${\displaystyle {}\varphi {|}_{U}=\operatorname {Id} _{U}}$ holds.