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hw6 - ECE130C Home Work 6 Two pages Due Spring 08 1 Reading...

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ECE130C Home Work 6 Spring ’08 Two pages. Due May 20, 2008. 1. Reading Assignment . Read Chapter 4 of the text book. Also browse through section 3.5 of the text book (it will not be included in the finals). 2. Suppose P is the projection matrix onto a line through a. (a) Why is the inner product of x with Py equal to the inner product of Px with y ? (b) Are the two angles the same? Find their cosines if a = (1 , 1 , 1) , x = ( - 2 , 0 , 1) , y = (2 , - 1 , 2). (c) Why is the inner product of Px with Py again the same? What is the angle between those two? 3. Construct the projection matrices P 1 and P 2 onto the lines through a ’s as defined below. Is it true that ( P 1 + P 2 ) 2 = P 1 + P 2 ? This would be true if P 1 P 2 = 0. a = (0 , 1) T for P 1 a = (1 , - 1) T for P 2 4. Project a 1 = (0 , 1) onto a 2 = (1 , - 2). Then project the result back onto a 1 . Draw these projections and multiply the projection matrices P 1 P 2 . Is this a projection? How about P 1 + P 2 ? 5. With b = 0 , 9 , 6 , 23 at t = 0 , 1 , 3 , 4 set up and solve the normal equations A T A ˆ x = A T b . For the best straight line as in Figure 3 . 9 a
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