Let $F$ be a field, let $V$ be a [vector space](/page/Vector%20Space) over $F$, and let $T: V \to V$ be an $F$-linear operator. Let $k \in \mathbb{N}$, and let $\lambda_1,\ldots,\lambda_k \in F$ be pairwise distinct eigenvalues of $T$. For each $i \in \{1,\ldots,k\}$, choose a nonzero vector $v_i \in E_{\lambda_i}(T)$. Then the vectors $v_1,\ldots,v_k$ are linearly independent over $F$.