φ(u+v) = λu + μv = λ’(u+v)
(λ- λ’)u + (μ- λ’)v = 0
u and v are linearly independent so λ = λ’ , μ = λ’
let be a linear transformation.
let be eigenvector of with the eigenvalue
let be eigenvector of with the eigenvalue
I need to prove that if is an eigenvector of , then
this is what I tried:
on the other hand:
but how can I conclude that and ?
Thanks for the help.
I know that vectors of different eigenvalues are linear independent, you don't need to convince me about that.
the thing is: it's not said that .
it's only said that is a eigenvector.
are you saying that I need to assume that and show that it leads to a contradiction?
(just want to wrap my head around it)