pset5 - 18.06 Problem Set 5 Due Wednesday 19 March 2008 at...

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18.06 Problem Set 5 Due Wednesday, 19 March 2008 at 4 pm in 2-106. Problem 1: Do problem 12 from section 4.3 (pg. 217) in the book. There is a typo in part c. It should read something like this: c) Let’s analyze what happens when b = (1 , 2 , 6). In this case x = 3 and the projection onto the line is p = (3 , 3 , 3). Check that p is perpendicular to e . Also find the projection matrix P . Problem 2: Do problem 17 from section 4.3 (pg. 217) in the book. Problem 3: Find a function of the form f ( t ) = C sin( t )+ D cos( t ) that approximates the three points (0 , 0), ( π/ 2 , 2), and ( π, 1). As explained in the book, the method is the same as for fitting a line using least-squares! (See pg. 212 for a quadratic example.) The difference is that the matrix A we use will no longer have columns with entries 1 and t i but rather sin( t i ) and cos( t i ). Problem 4: a) Show that if Q is orthogonal (i.e. Q is square with orthonormal columns) then so is Q T . Use the criteria Q T Q = I . b) If Q 1 and Q 2 are orthogonal, show that their product Q 1 Q 2 is as well.
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