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115apracticeexam2-sol

115apracticeexam2-sol - Practice Hour Exam#2 Solutions Math...

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Unformatted text preview: Practice Hour Exam #2 Solutions Math 115A Section 3 NAME: SOLUTIONS SCORES: 1. / 20 2. / 20 3. / 10 4. / 15 5. / 20 6. / 15 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 (20 points - 5 points each) (a) Suppose T : V → V is a linear operator. Define what it means for a vector from V to be an eigenvector of T . v ∈ V is an eigenvector of T if v 6 = ~ 0 and there is a scalar λ ∈ F such that T ( v ) = λ · v . (b) Suppose T : V → V is a linear operator and λ is an eigenvalue of T . Define the geometric multiplicity of λ . The geometric multiplicity of λ is the dimension of the eigenspace E λ ( T ) = N ( T- λI V ). (c) Define what it means for two n × n matrices to be similar . If A and B are n × n matrices, then A and B are similar if there exists an invertible n × n matrix Q such that B = Q- 1 AQ . (d) Define what it means for two vector spaces to be isomorphic . Vector spaces V and W are isomorphic if there exists an invertible linear trans- formation (or isomorphism) T : V → W . Problem 2 (20 points - 5 points each) For each of the following statements, determine if they are true or false. If they are true, prove them. If they arethem....
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115apracticeexam2-sol - Practice Hour Exam#2 Solutions Math...

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