115asamp4 - 1 Let V be a vector space with inner product(a...

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1. Let V be a vector space with inner product ( , ). (a) Show that if u, v V satisfy k u + v k = k u - v k then u and v are orthogonal. (b) Suppose that v, w V instead satisfy k v k = 1, k w k = 2, and k v + w k = 1. Find ( v, w ). 2. Let V be a vector space and { v 1 , ..., v k } be an orthogonal set of vectors in v . (a) Show that { v 1 , ..., v k } is linearly independent. (b) Let W = span { v 1 , ..., v k } . For v V write down formulas for Proj W ( v ) and Proj W ( v ). 3. Let V be an vector space with inner product ( , ). Deﬁne W and prove that dim( W ) = n - dim( W ) and that V = W W . You can use that any orthogonal basis for W can be extended to an orthogonal basis for V . 4. Let V = P 2 with inner product ( f, g ) = R 1 - 1 f ( x ) g ( x ) dx . Using the Gram-Schmidt process, ﬁnd an orthogonal basis for V . You can start with the basis
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