1. ## Eigenvalue proof

I need help with this proof

"T is a linear operator on a fin-dim vector space V
Beta is an ordered basis for V
Prove that Lambda is an eigenvalue of T iff Lambda is an eigenvalue of [T](subBeta)"

2. Originally Posted by mlemilys
I need help with this proof

"T is a linear operator on a fin-dim vector space V
Beta is an ordered basis for V
Prove that Lambda is an eigenvalue of T iff Lambda is an eigenvalue of [T](subBeta)"
Let $T:V\to V$ be a linear transformation with $k$ and eigenvalue, so $Tv = kv$ for some $v\in V^{\times}$.
Let $A = [T]_{\beta}$ be the corresponding matrix with respect to this ordered basis $\beta$.

Remember that $(Tv)_{\beta} = A(v)_{\beta}$ where $( ~ )_{\beta}$ is the coordinate with respect to $\beta$.
Since $Tv = kv \text{ iff } (Tv)_{\beta} = (kv)_{\beta} \text{ iff }A(v)_{\beta} = k(v)_{\beta}$.

Thus, $k$ is an eigenvalue of $A$ since $(v)_{\beta}$ is non-zero.