Vector space/Linear subspace/Projection/Definition

Linear Projection

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

$\varphi \colon V\longrightarrow V$

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