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10 Nov 2019
Full step-by-step solutions will be chosen.
Consider the function f : R3 times R3 rightarrow R defined by where b,c in R. Let Determine b and c such that Then either prove directly from the definition that f, using these values for b and c, is an inner product or give a counterexample to show that it is not. Let V be a finite-dimensional inner product space. Prove that for all Let P2(R) have the inner product p(x), q(x) = p(-1) + p(0)q(0) + p(1)q(1). Verify that S = {x2 + x + 1, x2 + 3x - 2, 19x2 - 3x - 14} is an orthogonal set under this inner product. Write each of 1, x and x2 as linear combinations of the elements of S. Let B and C be orthonormal bases of Rn. Prove that the change of coordinates matrixCPB is an orthogonal matrix. Let p1(x) = x, p2(x) = x2, and p3(x) = 1. Then {p1(x).p2(x),p3(x)} is a basis of P2(R). Use the Gram-Schmidt procedure on this basis, with the vectors in the given order, to find an orthonormal basis for P2(R) under the inner product p(x),q(x)) = p{0)q(0) + p(1)q(1) + p(2)q(2). Repeat part a) using the order p1(x) = x2, p2(x) = 1, and p3(x) = x.
Full step-by-step solutions will be chosen.
Consider the function f : R3 times R3 rightarrow R defined by where b,c in R. Let Determine b and c such that Then either prove directly from the definition that f, using these values for b and c, is an inner product or give a counterexample to show that it is not. Let V be a finite-dimensional inner product space. Prove that for all Let P2(R) have the inner product p(x), q(x) = p(-1) + p(0)q(0) + p(1)q(1). Verify that S = {x2 + x + 1, x2 + 3x - 2, 19x2 - 3x - 14} is an orthogonal set under this inner product. Write each of 1, x and x2 as linear combinations of the elements of S. Let B and C be orthonormal bases of Rn. Prove that the change of coordinates matrixCPB is an orthogonal matrix. Let p1(x) = x, p2(x) = x2, and p3(x) = 1. Then {p1(x).p2(x),p3(x)} is a basis of P2(R). Use the Gram-Schmidt procedure on this basis, with the vectors in the given order, to find an orthonormal basis for P2(R) under the inner product p(x),q(x)) = p{0)q(0) + p(1)q(1) + p(2)q(2). Repeat part a) using the order p1(x) = x2, p2(x) = 1, and p3(x) = x.
Nelly StrackeLv2
4 Feb 2019