Let T be a linear operator on a finite-dimensional vector space V over a field F, let 0

Chapter 5, Problem 13

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Let T be a linear operator on a finite-dimensional vector space V over a field F, let 0 be an ordered basis for V, and let A = [T]/3. In reference to Figure 5.1, prove the following. (a) If r G V and 0(v) is an eigenvector of A corresponding to the eigenvalue A, then v is an eigenvector of T corresponding to A. (b) If A is an eigenvalue of A (and hence of T), then a vector y G Fn is an eigenvector of A corresponding to A if and only if 4>~Q (y) is an eigenvector of T corresponding to A