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115hw5solutions

# 115hw5solutions - Math 115A Homework 5 Due December 5 2008...

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Math 115A Homework 5 Due December 5, 2008 1. Given the field F , the F -vector space V prove or disprove that the given function < , > : V × V F is an inner product. (2 pts each) a) F = R , V = R the space of sequences ( a n ) n N of real numbers such that a n = 0 for all but finitely many n , and < ( a n ) , ( b n ) > = X n =0 a n b n . This is easily seen to be an inner product. b) F = C , V = P 3 the space of complex polynomials of degree at most 3, and < f, g > = f (0) g (0) + f (1) g (1) + f (2) g (2) + f (3) g (3) . The linearity properties and conjugate symmetry are easy. It is positive definite because < f, f > is a sum of norm squares and a polynomial of degree at most 3 has at most 3 zeroes. 2. Let V be a finite-dimensional inner product space and let S : V V and T : V V be two self-adjoint linear transformations. Assume S T = T S . Show that S and T are simultaneously diagonalizable, that is, there exists a basis B of V such that each v ∈ B is an eigenvector for both S and T . (You may use the theorem that every self-adjoint transformation is diagonalizable.)

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115hw5solutions - Math 115A Homework 5 Due December 5 2008...

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