Why is $E_{\lambda}$ the kernel of the linear map $\alpha-\lambda I$

Question

The book starts the chapter on Eigenvalues and Eigenvectors, and goes that this statement is obvious. Here $E_{\lambda}$ stands for the set of vectors $v$ such that $α(v) = λv$, for any scalar $\lambda$.

Could somebody provide some intuition why is that obvious?

Thanks!

Answer 1

$\alpha(v) = \lambda v$ if and only if $\alpha(v) - \lambda v = 0$ if and only if $(\alpha - \lambda I)(v) = 0$.