Endomorphism/Eigenspaces are linear subspaces/Zero/Fact

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

$\phi :V⟶V$

a linear mapping. Then the following statements hold.

1. Every eigenspace

   ${\mathrm{Eig}}_{\lambda }\left(\phi \right)$

   is a linear subspace of $V$.

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

Proof, Write another proof