Projection/Linear subspace/Idempotent/Fact

direct sum decomposition

${\displaystyle {}V=U\oplus U'\,,}$

the projection onto ${\displaystyle {}U}$ is a projection in the sense of Definition. Such a projection

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

gives a decomposition

${\displaystyle {}V=\operatorname {kern} \varphi \oplus \operatorname {Im} \varphi \,,}$

${\displaystyle {}\varphi }$

${\displaystyle {}\operatorname {Im} \varphi }$