Proving that non-zero vectors are linearly independent given a 3x3 matrix

Question

I am having difficulty approaching this problem (below), I think it requires to construct $3x3$ matrices then using Gauss-Jordan Elimination to using the get the RREF. Although, I believe I am missing some steps. Any hints or tips would be greatly appreciated.

If C is a $3×3$ matrix and if the non-zero vectors $u$ and $v ∈ R^{3}$ are such that

$$Cu = 2u\ and \ Cv = −5v$$

show that $u$ and $v$ are linearly independent.

Answer 1

If they were linearly dependent, then there would be a $\lambda\in\mathbb R\setminus\{0\}$ such that $v=\lambda u$. Now, find a contradiction between this and what you know about $u$ and $v$.

Answer 2

Hint: start with $\alpha_1u+\alpha_2v=\bar0$ for $\alpha_1,\alpha_2\in F$ and apply $C$ to both sides. Use the facts about $Cu$ and $Cv$ to help conclude $\alpha_1=\alpha_2=0$.