Assignment 9

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MATH 223, Linear Algebra Winter, 2008 Assignment 9, due in class Wednesday, April 2, 2008 1. Let V be the vector space of real-valued functions defined and continuous on [ - π 2 , π 2 ] . (a) Show that, for any f V , ( Z π 2 - π 2 f ( x ) cos xdx ) 2 2 Z π 2 - π 2 f ( x ) 2 cos xdx. Identify those function f for which equality holds. (b) Show that, for any f V , ( Z π 2 - π 2 f ( x ) cos 2 xdx ) 2 4 3 Z π 2 - π 2 f ( x ) 2 cos xdx. Identify those function f for which equality holds. 2. (a) Suppose that V is any inner product space, and that ~v and ~w are orthogonal vectors in V . Show that || ~v + ~w || 2 = || ~v || 2 + || ~w || 2 . (b) Now suppose that V is finite-dimensional and W is a subspace of V . For any ~u V , show that Proj W ~u is the "closest" vector to ~u that's in W ; that is, if ~x = Proj W ~u and ~ y 6 = ~x but ~ y W , then || ~u - ~x || < || ~u - ~ y || . 3. Let V = C 5 with the standard inner product. Let W = Span 1 1 0 1 - i 1 + i , 2 - 4 i 4 1 1 - 6 i 4 + i , - 2 + 4 i 6 - 2 i 6 - 2 i 2 2 + 4 i . (a) Find an orthonormal basis for each of W and W . (b) Find Proj W 1 0 i 0 1 . 4. For each of the following Hermitian matrices A , find a unitary matrix U such that U T AU is diagonal, and find the diagonal matrix. A = 3 2 2 2 3 2 2 2 3 , A = 0 1 + i 2 + i 1 - i 1 3 - i 2 - i 3 + i 4 . 1
[Note that the first of these is in fact a real symmetric matrix, so that U can be chosen orthogonal.] 5. (a) Suppose that A is a symmetric matrix over the reals with nonnegative eigenvalues. Show that there is a symmetric real matrix B such that B 2 = A . [Such a matrix is called a square root of A .] (b) Find a symmetric square root of the matrix 2 1 1 1 2 1 1 1 2 . 6. Suppose that ( a 1 , b 1 ),. . . ,( a n , b n ) are points in the plane R 2 . If n 3, it's unlikely that there is a line going through all of them. (a) Show that the line defined by y = αx + β goes through all of them if and only if α β is a solution to A~x = ~ b , where A = a 1 1 a 2 1 . . . a n 1 and ~ b = b 1 b 2 . . . b n . As mentioned, this is usually impossible if n 3. But the system A T A~x = A T ~ b frequently has a unique solution. (b) Show that, if A is as above, and not all the values a 1 ,. . . , a n are the same, then A T A has positive determinant and hence there is a unique solution to A T A~x = A T ~ b . The solution α β is known as the least squares solution to the original system A~x = ~ b , and the line defined by y = αx + β is the line of best fit (to the data ( a 1 , b 1 ),. . . ,( a n , b n )). (c) If n = 4 and ( a 1 , b 1 ) = (0 , 2), ( a 2 , b 2 ) = (1 , 3), ( a 3 , b 3 ) = (2 , 5) and ( a 4 , b 4 ) = (3 , 6) find the least-squares solution to A~x = ~ b . 2
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