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.

1. Every eigenspace

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

   is a linear subspace of ${}V$.

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

Proof, Write another proof