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!
Suppose u and v are not independent. That is, suppose there exist numbers, A and B, not both 0 such that Au+ Bv= 0. Then C(Au+ Bv)= ACu+ BCv= A(2u)+ B(-5v)= 2Au- 5Bv= 2(Au+ Bv)- 3Bv= -3Bv= 0.  It follows that v= 0, a contradiction.