linalghw6 - 1 Homework 6 1. let V = R 2 . none of the the...

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Unformatted text preview: 1 Homework 6 1. let V = R 2 . none of the the following are inner-products. for each one, give, with explanation, an axiom which is violated. a. ( u,v ) = u 1 v 1- u 2 v 2 b. ( u,v ) = u 1 v 1 + u 2 v 1 + u 2 v 2 c. ( u,v ) = 0 d. ( u,v ) = u 1 v 2- 5. 2. let V = C 3 , with the inner-product ( u,v ) = u 1 v 1 + 2 u 2 v 2 + u 3 v 3 , and let U = span { (1 , , 1) , (1 ,i, 0) } V . find an orthonormal basis of U using gram-schmidt. 3. let V = f : [0 , 2 ] R f is continuous , the vector space of all continuous real-valued functions on the interval [0 , 2 ], and define an inner-product by ( f,g ) = Z 2 f ( x ) g ( x ) dx. let U be the three-dimensional subspace spanned by U = span { 1 , sin , cos } . a. show that { 1 , sin , cos } are an orthogonal basis of U . find and orthonormal basis. b. let f V be the function f ( x ) = x x 2 - x < x 2 1 which looks like a tent. find the projection proj U ( f ) of f onto U , and plot a couple points. is it pretty close tocouple points....
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linalghw6 - 1 Homework 6 1. let V = R 2 . none of the the...

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