MA554_AUG99 - Math 554 Heinzer Qualifying Exam August 1999 Name(20 1 Let A ∈ C 4 × 4 be a diagonal matrix with main diagonal entries 1 2 3 4

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Unformatted text preview: Math 554 - Heinzer Qualifying Exam August 1999 Name: (20) 1. Let A ∈ C 4 × 4 be a diagonal matrix with main diagonal entries 1 , 2 , 3 , 4. Define T A : C 4 × 4 → C 4 × 4 by T A ( B ) = AB- BA . (i) What is dim(ker( T A ))? (ii) What is dim(im( T A ))? (iii) What are the eigenvalues of T A ? (iv) What is the minimal polynomial of T A ? (v) Is T A diagonalizable? Explain. (12) 2. (i) Let A ∈ Z 3 × 4 and define φ A : Z 4 → Z 3 by φ A ( X ) = AX . True or False? If φ A is surjective, then the determinant of some 3 × 3 minor of A is a unit of Z . Explain. (ii) Let B ∈ Z 4 × 3 and define φ B : Z 3 → Z 4 by φ B ( X ) = BX . True or False? If the determinant of some 3 × 3 minor of B is nonzero, then φ B is injective. Explain. (10) 3. True or False? If A ∈ R n × n is normal and if the eigenvalues of A are all real, then A is symmetric. Justify your answer. (12) 4. Let V be a vector space over an infinite field F . Prove that V is not the union of finitely many proper subspaces.union of finitely many proper subspaces....
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MA554_AUG99 - Math 554 Heinzer Qualifying Exam August 1999 Name(20 1 Let A ∈ C 4 × 4 be a diagonal matrix with main diagonal entries 1 2 3 4

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