4) Let A E Mnxn(F) be upper triangular. Prove that the diagonal entries of A are the eigenvalues of A. For an eigenvalue A of A, prove that ma(A) is equal to the number of times A appears on the diagonal of A. 5) Let V be finite-dimensional. Let T : V → V be a linear operator on V. Let W C V be a T-invariant subspace. Suppose T is diagonalizable with (distinct) eigenvalues A1, -.· , Ak. Prove that W = O=(WN Ex,). k i=1

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