10. Let T be a linear operator on a finite-dimensional vector space V with the distinct...

 

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10. Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues A₁, A2,..., Ak and corresponding multiplicities m1, m2,...,mk. Suppose that 3 is a basis for V such that [T] is an upper triangular matrix. Prove that the diagonal entries of [T] are A1, A2,..., Ak and that each X, occurs m; times (1 ≤ i ≤k). 10. Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues A₁, A2,..., Ak and corresponding multiplicities m1, m2,...,mk. Suppose that 3 is a basis for V such that [T] is an upper triangular matrix. Prove that the diagonal entries of [T] are A1, A2,..., Ak and that each X, occurs m; times (1 ≤ i ≤k).

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