1. True or false? If the triangular matrix A is similar to a diagonal matrix, then A is already diagonal.

2. Let T be a linear operator on a finite-dimensional vector space over an algebraically closed field F. Let f be a polynomial over F. Prove that c is a characteristic value of f(T) if and only if c = f(t), where t is a characteristic value of T.

3. Let V be the space of n x n matrices over F. Let A be a fixed n x n matrix over F. Let T and U be the linear operators on V defined by T(B) = AB; U(B) = AB - BA. a) True or false? If A is diagonalizable (over F), then T is diagonalizable. b) True or false? If A is diagonalizable, then U is diagonalizable.

2. Let T be a linear operator on a finite-dimensional vector space over an algebraically closed field F. Let f be a polynomial over F. Prove that c is a characteristic value of f(T) if and only if c = f(t), where t is a characteristic value of T.

3. Let V be the space of n x n matrices over F. Let A be a fixed n x n matrix over F. Let T and U be the linear operators on V defined by T(B) = AB; U(B) = AB - BA. a) True or false? If A is diagonalizable (over F), then T is diagonalizable. b) True or false? If A is diagonalizable, then U is diagonalizable.

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