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h11-3 - subspace of U 5 Let V be the vector space over the...

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COT5615 Mathematics for Intelligent Systems Fall 2011 Home Work Assignment 3: Due Monday 10/31/11 before class In the problems below, the matrices A = 1 2 3 4 5 7 6 8 9 B = 1 2 3 4 5 6 1. Hand compute the LU decomposition of A 2. Hand compute the QR decomposition of A . 3. Invert A using the solution of the first question above. 4. Considering B : U V as a linear transform, find the subset of U that maps to the zero vector in V . Prove that this subset is a subspace of U . 5. Let
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Unformatted text preview: subspace of U . 5. Let V be the vector space over the complex numbers of all functions from R into C , i.e., the space of all complex-valued functions on the real line. Let f 1 ( x ) = 1 ,f 2 ( x ) = e ix ,f 3 ( x ) = e-ix . Prove that f 1 ,f 2 ,f 3 are linearly independent. Let g 1 ( x ) = 1 ,g 2 ( x ) = cos x,g 3 ( x ) = sin x . Find an invertible 3 × 3 matrix P such that g j = 3 X i =1 P ij f i . 1...
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