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?
the general case is much more interesting: suppose are eigenvectors with corresponding pairwise distinct eigenvalues then are linearly independent:
the proof is by induction over : there's nothing to prove for n = 1. so suppose the claim is true for and let for some scalars call this (1). we want to prove that
: from (1) we have call this (2). now multiply (1) by and then subtract the result from (2)
to get: hence by induction hypothesis we must have which gives us because are assumed
to be pairwise distinct. so by (1). this completes the proof.