Eigenvalues/Endomorphism/Matrix/Fact

Let ${}\varphi \colon V\rightarrow V$ be an endomorphism on the finite-dimensional ${}K$-vector space ${}V$, and let ${}{\mathfrak {u}}=u_{1},\ldots ,u_{n}$ denote a basis of ${}V$. Let ${}M=M_{\mathfrak {u}}^{\mathfrak {u}}$ be the describing matrix of ${}\varphi$ with respect to the basis.

Then

${}v\in V$ is an eigenvector of ${}\varphi$ for the eigenvalue ${}a$ if and only if the coordinate tuple

of ${}v$ with respect to the basis is an eigenvector of ${}M$ for the eigenvalue ${}a$.

Proof, Write another proof