To give a little bit more detail on 0-0-0's answer: Let $V$ be an $\mathbb{F}$-vector space of finite dimension.
We first prove the following theorem:
Theorem: Let $\phi\in\mathsf{Hom}(V,V)$, $U\subseteq V$ invariant under $\phi$. Let $\phi|_U:U\to U$ be the restriction of $\phi$ to $U$. Then $q_{\phi|_U}$ divides $q_\phi$ where $q_\phi$ is the minimal polynomial of $\phi$, similar for $\phi|_U$.
Proof: Let $q_\phi$ be the minimal polynomial, i.e. $q_\phi(\phi)=\mathbf{0}$ where $\mathbf{0}$ is the null-endomorphism. Now still $q_\phi(\phi)|_U=\mathbf{0}$. It is straightforward to check, as $U$ is invariant, that $q_\phi(\phi)|_U=q_\phi(\phi|_U)=\mathbf{0}$. Now, per definition $q_{\phi|_U}(\phi|_U)=\mathbf{0}$ and as this minimal polynomial generates the ideal
$$I(\phi|_U)=\{p\in\mathbb{F}[X]\mid p(\phi|_U)=\mathbf{0}\}$$
we have $q_{\phi|_U}\mid q_\phi$. $\Box$
As a corollary, we obtain the following:
Corollary: Let $\phi\in\mathsf{Hom}(V,V)$. If $\phi$ is diagonalizable and $U\subseteq V$ is invariant under $\phi$, then $\phi|_U:U\to U$ is diagonalizable.
Proof: $\phi$ is diagonalizable iff $$q_\phi=\prod_{i=1}^m(X-\lambda_i)$$ for pairwise distinct eigenvalues $\lambda_i$ of $\phi$. Now, by the above theorem, we have that $q_{\phi|_U}$ divides $q_\phi$, i.e. also $q_{\phi|_U}$ splits into distinct linear factors with algebraic multiplicity 1. Thus $\phi|_U$ is diagonalizable. $\Box$
Now let $U\subseteq V$ be a $\phi$-invariant subspace for $\phi\in\mathsf{Hom}(V,V)$ and $\mathrm{dim}(U)=k$. By the corollary, we have that $\phi|_U$ is diagonalizable, i.e. there exists an eigenbasis $B=(v_1,\dots,v_k)$ for $U$ s.t. $\phi|_U$ is represented by a diagonal matrix. Now, the vectors in this eigenbasis for $U$ are obviously also eigenvector w.r.t. $\phi$. Thus $U=\mathrm{span}(v_1,\dots,v_k)$ for these basis vectors as desired.
We can summarize this good with the following theorem:
Theorem: Let $\phi\in\mathsf{Hom}(V,V)$ be diagonalizable. Then a subspace $U\subseteq V$ with $\mathrm{dim}(U)=k$ is $\phi$-invariant iff $U=\mathrm{span}(v_1,\dots,v_k)$ for eigenvectors $v_i$.
The one direction we just proved, the other is a simple observation.
