Exercise 5.A.31

Proof. Suppose $v_1,\dots ,v_m$ is linearly independent. Let $T\in L(V)$ be any linear map such that

for some distinct ${\lambda }_1,\dots ,{\lambda }_m\in 𝔽$. Therefore $v_1,\dots ,v_m$ are eigenvectors of $T$ corresponding to distinct eigenvalues.

Converse is the same as 5.10. □