asst7 - MATH 235 Fall 2007 Assignment#7 1 Let V= C 2 and...

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Unformatted text preview: MATH 235, Fall 2007 Assignment #7 1. Let V= C 2 and for all x,y ∈ V define h x,y i = x T A y, where A = • 1 i- i 2 ‚ . (a) Prove that h· , ·i defines an inner product on V . (b) In this inner product space, compute h x,y i for x = • 1- i 2 + 3 i ‚ , and y = • 2 + i 3- 2 i ‚ . 2. Provide reasons why each of the following is not an inner product on the given vector spaces V . (a) V = R 2 , h ( a,b ) , ( c,d ) i := ac- bd. (b) V = R 2 × 2 , h A,B i := trace ( A + B ) . (c) V = P ( R ) (all real-valued polynomials), h f,g i := R 1 f ( t ) g ( t ), where denotes differentiation. 3. (a) Let S = { (1 ,i, , 1) , (1- i, 2 , 4 i, 1) , (3 + i, 4 i,- 4 , 1) } . Find an orthonormal basis for Span ( S ). (b) Let x = (1 , 1 , 1 , 1). Note that Span ( S ) is a subspace of C 4 , and x ∈ C 4 . Use the othonormal basis you found in (a) to express x as b y + z , where b y ∈ Span ( S ) and z ∈ Span ( S ) ⊥ ....
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asst7 - MATH 235 Fall 2007 Assignment#7 1 Let V= C 2 and...

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