Projection/Linear subspace/Idempotent/Fact

Let ${}K$ be a field and ${}V$ a ${}K$-vector space.

For a

direct sum decomposition

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

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

$\varphi \colon V\longrightarrow V$

gives a decomposition

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

and ${}\varphi$ is the projection onto ${}\operatorname {Im} \varphi$.

Proof, Write another proof