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test4-sp94

# test4-sp94 - 1 T-1 1 2 T A = 1-1 1 1 1 1 2 b = 2 4(a Find...

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MAS 4105 Test 4 March 22, 1994 Show all your work on the paper provided. Your work should be written in a proper and coherent manner. 50 points total. 1. (14 points) (a) Complete the following definition. An inner product on a vector space V is an operation on V that assigns to each pair of vectors x and y in V a real number h x , y i satisfying the following conditions: (b) Prove that in an inner product space V , h 0 , v i = 0 for every v in V . (c) Let S be a subspace of an inner product space V and define S := { x V | h x , y i = 0 for every y S } . Prove that S is a subspace of V . 2. (6 points) Let x = ( - 3 , 1 , - 1 , 2) T . Find the following norms: (a) || x || 1 (b) || x || 2 3. (4 points) Sketch the set of points ( x 1 , x 2 ) = x T in R 2 such that || x || = 1 . 4. (16 points) Let S = Span ((1
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Unformatted text preview: , 1) T , (-1 , , 1 , 2) T ), A = 1-1 1 1 1 1 2 , b = 2 4 . (a) Find bases for S and S ⊥ . (b) Solve the normal equations A T A x = A T b . (c) Find the orthogonal projection p of b onto S (ie. ﬁnd the vector p ∈ S such that b-p ∈ S ⊥ ). (d) Find min s ∈ S || b-s || ; ie. ﬁnd the minimum of || b-s || for s ∈ S and b = (0 , 2 , , 4) T . 5. (10 points) Let U and V be subspaces of a vector space W . (a) Deﬁne what it means to say that W is the direct sum of U and V . (b) Prove that if W = U ⊕ V then U ∩ V = { } . 1...
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