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**Unformatted text preview: **Math 425, Spring 2011 Jerry L. Kazdan Problem Set 6 DUE: Thursday March 17 [ Late papers will be accepted until 1:00 PM Friday ]. 1. This problem concerns orthogonal projections into a subspace of a larger space. a) Let U = ( 1 , 1 , , 1 ) and V = (- 1 , 2 , 1 ,- 1 ) be given orthogonal vectors in R 4 and let S be the two dimensional subspace they span. Write the vector X = ( 1 , 1 , 1 ,- 1 ) in the form X = a U + b V + W , where a , b are scalars and W is a vector perpendicular to U and V . We call X 1 = a U + b V , which is in S , the orthogonal projection of X into S . The notation P S X = a U + b V for the projection of X into S is sometimes helpful. b) Let T be the three dimensional space spanned by U , V , and W . Find the orthogonal projection of Y = ( 1 , , , ) into T . c) Find an orthonormal basis for S and then for T . 2. Suppose f is a function of one variable that has a continuous second derivative. Show that for any constants a and b , the function u ( x , y ) : = f ( ax +...

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