20095ee103_1_hw2_sols

20095ee103_1_hw2_sols - L. Vandenberghe 10/8/09 EE103...

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Unformatted text preview: L. Vandenberghe 10/8/09 EE103 Homework 2 solutions 1. Exercise 2.3. (a) Ax is a circular downward shift over one position: Ax = 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 x 1 x 2 x 3 x 4 x 5 = x 5 x 1 x 2 x 3 x 4 . (b) A k is a circular shift over k positions. A 2 = 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 , A 3 = 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 , A 4 = 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 , and A 5 = I . (If we shift x over five positions, we obtain x again.) (c) A T x shifts x in the other direction: A T x = 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 x 1 x 2 x 3 x 4 x 5 = x 2 x 3 x 3 x 4 x 1 . A T is also the inverse of A , because A T A = I . 2. Exercise 2.5. f is linear. x T y is a scalar, so ( x T y ) y = y ( x T y ) = y ( y T x ) = ( yy T ) x....
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This note was uploaded on 05/10/2010 for the course EE EE103 taught by Professor Jacobson during the Spring '09 term at UCLA.

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20095ee103_1_hw2_sols - L. Vandenberghe 10/8/09 EE103...

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