Endomorphism/Eigenspaces are linear subspaces/Zero/Fact/Proof

Proof

(1). Due to ${}\varphi (0)=0=\lambda 0$, the zero vector belongs to ${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}$. Let ${}u,v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}$ and let ${}w=au+bv$. Then

${}\varphi (w)=a\varphi (u)+b\varphi (v)=a\lambda u+b\lambda v=\lambda (au+bv)=\lambda w\,.$

(2) and (3) follow directly from the definitions.

To fact