Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v.
The Attempt at a Solution
Hey everyone I've been doing sample questions in the build up to an exam and I came across this. Any help would be greatly appreciated as I'm struggling a bit.
Here is what I know:
- λ is an eigenvalue if there exists a non-zero vector v∈V such that Tv = λv.
- I also read this for complex: q(λ) = det (λI - T), where the zeros of q(λ) in ℂ are the eigenvalues of T.
What does the second point mean or how would I answer this properly. Thanks in advance.