(a) Let T be a linear operator on a vector space V over the field F, and let g(t) be a
Chapter 5, Problem 22(choose chapter or problem)
(a) Let T be a linear operator on a vector space V over the field F, and let g(t) be a polynomial with coefficients from F. Prove that if x is an eigenvector of T with corresponding eigenvalue A, then g(T)(x) = g(X)x. That is, x is an eigenvector of o(T) with corresponding eigenvalue g(X). (b) State and prove a comparable result for matrices. (c) Verify (b) for the matrix A in Exercise 3(a) with polynomial g(t) = 2t2 t + 1, eigenvector x = I J, and corresponding eigenvalue A = 4.
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