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115apracticefinal-sol

# 115apracticefinal-sol - Practice Final Exam Math 115A NAME...

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Practice Final Exam Math 115A NAME: SOLUTIONS SCORES: 1. / 20 7. / 10 2. / 20 8. / 20 3. / 20 9. / 20 4. / 15 10. / 20 5. / 25 11. / 20 6. / 10 Total: / 200 SIGNATURE: I hereby swear that in taking this exam I have adhered to all of the university’s rules on academic integrity.

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Problem 1 (20 points - 5 points each) (a) Suppose that V and W are vector spaces over the same field F . Define L ( V, W ). L ( V, W ) is the vector space of all linear transformations T : V W . (b) Suppose T : V V is a linear operator and λ is an eigenvalue of T . Define the algebraic multiplicity of λ . The algebraic multiplicity of λ is the largest d such that ( t - λ ) d is a factor of the characteristic polynomial of T . (c) Define the Frobenius inner product on the vector space M n × n ( F ). For A, B M n × n ( F ), we have h A, B i = tr( B * A ). (d) Suppose V is an inner product space and S is a nonempty subset of V . Define the orthogonal complement S of S . S = { x V : h x, s i = 0 for all s S } .
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 are false, provide a counterexample . (a) If A and B are similar n × n matrices, then A and B have the same eigen- values. True: We have that A and B have the same characteristic polynomials; this was a homework problem. Then since the eigenvalues are the roots of the characteristic polynomial, we have that A and B have the same eigenvalues. (b) If V is an inner product space and x, y, z V are such that h x, z i = h y, z i , then x = y . False: Let V = R 2 , x = (1 , 0), y = (0 , 1), z = (0 , 0). Then h x, z i = h y, z i = 0 while x 6 = y .

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(c) If T : V W and U : W Z are linear and U T is onto, then T is onto. False: Let T : R R 2 be given by T ( x ) = ( x, 0) and U : R 2 R be given by U ( a, b ) = a . Then U T is the identity on R , which is onto, while T is not onto. (d) Suppose that V is a finite-dimensional vector space and S 1 and S 2 are subsets of V . Suppose that S 1 is linearly independent and S 2 spans V . Then S 1 cannot have more vectors in it than S 2 . True: The size of S 1 is at most dim( V ) and dim( V ) is at most the size of S 2 .
Problem 3 (20 points-5 points each) The following propositions are not stated correctly. By slightly modifying the proposition (e.g. adding or deleting a word), make them correct.

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115apracticefinal-sol - Practice Final Exam Math 115A NAME...

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