# hw5 - Math 425, Spring 2011 Jerry L. Kazdan Problem Set 5 D...

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Math 425, Spring 2011 Jerry L. Kazdan Problem Set 5 DUE: Thurs. Feb. 24 Late papers will be accepted until 1:00 PM Friday . 1. In R 4 the vectors U 1 : = ( 1 , 1 , 1 , 1 ) , U 2 : = ( 1 , 1 , - 1 , - 1 ) , U 3 : = ( 2 , - 2 , 2 , - 2 ) , U 4 : = ( 1 , - 1 , - 1 , 1 ) are orthogonal, as you can easily verify. a) Use these to ﬁnd an orthonormal basis e k : = α k U k , k = 1 ,..., 4. b) Write the vector v : = ( 0 , - 2 , 2 , 5 ) using this basis: v = a 1 e 1 + a 2 e 2 + a 3 e 3 + a 4 e 4 . c) Find the projection, Pv , of v into the plane spanned by U 2 and U 3 . d) Compute k Pv k . 2. Let X be a linear space with an inner product (not necessarily R n ) and let P : X X be an orthogonal projection , so P 2 = P and P = P * . Write V for the image of P ; it is the space into which vectors are projected. Given x X , write x = v + w , where v = Px is the projection of x into V . Show that w is orthogonal to V . 3. Let f ( x ) be a 2 π periodic function. Use Fourier series to investigate ﬁnding 2 π periodic solutions of - u 00 ( x )+ u = f ( x ) , so we want u and all of its derivatives to be 2 π periodic. This is routine – and short. Expand f in a Fourier series, so f ( x ) = k = - a k e ikx and seek the solution as a Fourier series u

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hw5 - Math 425, Spring 2011 Jerry L. Kazdan Problem Set 5 D...

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