Why different eigenvalues of a real matrix are associated with linear independent values?

Question

If we have a real matrix $A$ such that $AV=aV$ and $AW=bW$ for two different values $a$ and $b$ and such that $V$ and $W$ are not $0$, how can we prove that $V$ and $W$ are linearly independent?

Answer 1

Assume $W=kV\ne 0.$ Thus

$$0=A0=A(W-kV)=AW-kAV=bW-kaV=(b-a)W.$$ This is not possible if $a\ne b.$

So $V$ and $W$ are linearly independent.