Proof about eigenvalues/vectors of a linear transformation

I had to say whether the following affirmation was true or not and explain. I fell in love with the proof I gave : (I hope there's no flaws in it)

Let be a linear transformation and let and be 2 eigenvectors of with eigenvalues and respectively. If is linear independent.

I say true : Proof :

I have to prove , so I'll prove which is equivalent. In other words I have to prove that linear dependent implies .

So we have linear dependent such that .

We have that .

Is there any flaw?