How can I prove that if I have n eigenvectors from different eigenvalues, they are all linearly independent?
I’ll do it with two vectors. I’ll leave it to you do it in general.
Suppose v1 and v2 correspond to distinct eigenvalues λ1 and λ2, respectively.
Take a linear combination that is equal to 0, α1v1+α2v2=0. We need to show that α1=α2=0.
Applying T to both sides, we get
Now, instead, multiply the original equation by λ1:
Now take the two equations,
and taking the difference, we get:
Since λ2−λ1≠0, and since v2≠0 (because v2 is an eigenvector), then α2=0. Using this on the original linear combination 0=α1v1+α2v2, we conclude that α1=0 as well (since v1≠0).
So v1 and v2 are linearly independent.
Now try using induction on n for the general case.