A5_235_ONLINE_s2018.pdf - Math 235 Online Assignment 5 1 1...

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Math 235 Online Assignment 5 1. In this question, we are using the standard inner product on R 3 . Let S = Span be a subspace of R 3 and let ~w = 1 1 2 . a) Find an orthonormal basis for S . b) Find proj S ~w . c) Find a basis for S . 2. Suppose that P 2 ( R ) has inner product h , i that satisfies the following: h 1 , 1 i = 2 , h 1 , x i = 2 , 1 , x 2 = - 2 h x, x i = 4 , x, x 2 = - 2 , x 2 , x 2 = 3 Let B = { 2 , x, x 2 } . Apply the Gram-Schmidt procedure to B (in this order) to find an orthonormal basis C of P 2 ( R ). 3. Define the inner product h ~x, ~ y i = 2 x 1 y 1 + 2 x 2 y 2 + x 3 y 3 on R 3 . Extend the set 1 1 1 to an orthogonal basis for R 3 . 4. Let V be a finite-dimensional inner product space and let W be a subspace of V . Consider the linear map proj W : V V . Prove that Range(proj W ) = W . Furthermore, if W 6 = V (i.e. if dim W < dim V 1 - 1 1 , 1 0 1

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