Eigenspace is a Subspace

Theorem

For any $\lambda \in \mathbb{R}$, we have $E_{\lambda }$ is a subspace of $V$

Proof

1. Check that $E_{\lambda }$ is non-empty
2. Note that we have $T(0)=\overrightarrow{0}=\lambda 0$
3. Thus, $\overrightarrow{0}\in E_{\lambda }$ for any $\lambda \in \mathbb{R}$
4. Note that $0$ is not an Eigenvector, but it is in $E_{\lambda }$
5. Pick $x,y\in E_{\lambda }$, and $c\in \mathbb{R}$
6. Then, we get: $T(cx+y)$
7. $=cT(x)+T(y)$
8. $=c(\lambda x)+\lambda y$
9. $=\lambda (cx+y)$
10. Thus, $cx+y\in E_{\lambda }$
11. It follows that $E_{\lambda }$ is a Subspace