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115as11exam2-sol - Hour Exam#2 Math 115A Section 3 NAME...

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Unformatted text preview: Hour Exam #2 Math 115A Section 3 May 16, 2011 NAME: SOLUTIONS Bruin ID: SCORES: 1. / 15 2. / 15 3. / 30 4. / 20 5. / 20 Total: / 100 SIGNATURE: I hereby swear that in taking this exam I have adhered to all of the university’s rules on academic integrity. Problem 1 (15 points - 5 points each) (a) Suppose T : V → V is a linear operator, where V is a vector space over the field F . Define what it means for λ ∈ F to be an eigenvalue of T . λ is an eigenvalue of T if and only if there is a nonzero vector v ∈ V such that T ( v ) = λv . (b) Suppose that T : V → V is a linear operator and V is finite-dimensional. Define the characteristic polynomial of T . Let β be any basis for V and let A = [ T ] β . Then the characteristic polynomial of T is the polynomial p ( λ ) = det( A- λI ). (c) Let A be an m × n matrix over the field F . Define the linear transformation L A . L A : F n → F m is given by L A ( ~x ) = A · ~x for all ~x ∈ F n . Problem 2 (15 points - 5 points each) For each of the following statements, determine if they are true or false. If they are true,determine if they are true or false....
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115as11exam2-sol - Hour Exam#2 Math 115A Section 3 NAME...

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