I am having difficulty approaching this problem (below), I think it requires to construct $3×3$ 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.
Best Answer
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 $v$.