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Math Help - Eigenvalue proof

  1. #1
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    Talking 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)"
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  2. #2
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    Quote Originally Posted by mlemilys View Post
    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.
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