Eigenvectors with distinct eigenvalues are linearly independent

#linear_algebra

Suppose $A$ is a $5 \times 5$ matrix and $v_{1}, v_{2}, v_{3}$ are eigenvectors of $A$ with distinct eigenvalues. Prove ${v_{1}, v_{2}, v_{3}}$ is a linearly independent set. Hint: Consider a minimal linear dependence relation.

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Suppose $A$ is a $5\times 5$ matrix and $v_1,v_2, v_3$ are eigenvectors of $A$ with distinct eigenvalues. Prove $\{v_1,v_2,v_3\}$ is a linearly independent set. {\itshape Hint:} Consider a minimal linear dependence relation.