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assig4 - MATH 245 Linear Algebra 2 Assignment 4 Due Wed Nov...

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MATH 245 Linear Algebra 2, Assignment 4 Due Wed Nov 10 1: (a) Let A 1 = 1 2 - 1 0 , A 2 = 3 4 - 1 2 and A 3 = 3 - 1 - 2 4 . Apply the Gram-Schmidt Procedure to the basis U = A 1 , A 2 , A 3 to obtain an orthonormal basis for U = Span A 1 , A 2 , A 3 M 2 × 2 . (b) Find an orthonormal basis for P 2 with the inner product given by p, q = p (0) q (0) + p (1) q (1) + p (2) q (2) by applying the Gram-Schmidt Procedure to the standard basis 1 , x, x 2 . 2: Consider P 2 as a subspace of C [ - 1 , 1] with its standard inner product, which is given by f, g = Z 1 - 1 fg . Applying the Gram-Schmidt Procedure to the standard basis 1 , x, x 2 for P 2 yields the orthogonal basis p 0 , p 1 , p 2 , where p 0 = 1, p 1 = x and p 2 = x 2 - 1 3 . (a) Find the polynomial f P 2 which minimizes Z 1 - 1 ( f ( x ) - | x | ) 2 dx . (b) Let f ∈ C [ - 1 , 1]. Given that 1 = Z 1 - 1 f ( x ) dx = Z 1 - 1 xf ( x ) dx = Z 1 - 1 x 2 f ( x ) dx , find the minimum possible value for Z 1 - 1 f ( x ) 2 dx . 3: (a) Let U and V be inner product spaces over C . Let L : U V be a linear map, and suppose that the adjoint L * : V U exists. Show that Null( L * L ) = Null( L ) = Range(L * ) .
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