Endomorphism/Eigenspaces are linear subspaces/Zero/Fact

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

\varphi \colon V\longrightarrow V

a linear mapping. Then the following statements hold.

Every eigenspace
$\operatorname {Eig} _{\lambda }{\left(\varphi \right)}$

is a linear subspace of ${}V$ .
${}\lambda$ is an eigenvalue for ${}\varphi$ , if and only if the eigenspace ${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}$ is not the null space.
A vector ${}v\in V,\,v\neq 0$ , is an eigenvector for ${}\lambda$ , if and only if ${}v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}$ .