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length.”. ) 6 . C onsider this m athematical (and not necessarily computer) procedure: [ , v '] = project [u, v]
Inputs vectors u and v , computes and r eturns u v / u u and v ' v u.
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9 3 , u 2 5 , u 3 3 .
N ow , let u 0 1
3 1 a. P erform [r1,2 , u2 '] = project [u1 , u2 ] .(T hat is, subtract the projection of u2 onto the
subspace spanned by u1 .)
b. P erform [r1,3 , u3 '] = project [u1 , u3 ] .(T hat is, subtract the projection of u3 onto the
subspace spanned by u1 .)
c. P erform [r2,3 , u3 "] = project [u2 ', u3 '] .(T hat is, subtract the projection of u3 ' onto
the subspace spanned by u1 .) 1 r1,2 r1,3 d. C ompute A u1 u2 ' u3 " 0 1 r2,3 . (Compare to U in Problem 1. Y ou have
0 0 1 just used the Gram  Schmidt A lgorithm to orthogonalize – but not ortho normalize v ectors. T hat is, the normalizations are not done. )
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7 . a. G iven A 2 3 , use the Gram Schmidt algorithm to express A QR, w here QT Q I 1 2 and R is upper triangular.
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b. U se this to solve the least squares problem min Ax b , w here b 0 . x
1 H int: A smart student w ould now t est AT r , w here r Ax b. 8 . A nsw er true or false . If false offer a simple counterexample.
a. If the problem Ax b has a solution x, then the problem HAx Hb has the same
solution x, for any matrix H (for w hich HA and Hb are defined.)
b. L et matrix A have columns A.,1 solution x, then b x1 A.,1 ... xn A.,n . A.,n and the problem Ax b has a 9 . P rove that if yT x 0, for all x, then y 0. 1 0 . P rove that if QT Q I , then if x is perpendicular to y, then Qx is perpendicular to Qy....
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This homework help was uploaded on 03/21/2014 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas at Austin.
 Spring '08
 PAVLOVIC
 Matrices

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