How to show linear independency of non-zero vectors of a 3x3 matrix? If $C$ is a $3×3$ matrix and if the non-zero vectors $u$ and $v\in \Bbb R^3$ are such that, $Cu=2u$ and $Cv=−5v$ How can you show that $u$ and $v$ are linearly independent?

Suppose it is dependent. Then $$u=kv$$ where $k$ is non zero. Then $$Cu=kCv \implies 2u=-5kv$$ Now multiply $2$ on both sides of $u=kv$. Then we have $$2u=2kv$$

Subtract the two equations, we get $$0=7kv$$ and so $k=0$, a contradiction!