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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 18085 FALL 2002 EXAM 1 SOLUTIONS DETAILED SOLUTIONS Problem 1 Let x = (tl, ..., t,) and b = (bl, ..., b,). Then and we would like to solve (but can't) The short answer is that ATA is always positive definite. The long answer is that A is of rank 1only if all tl = ... = t, (all measurements taken at the same time and the m points lie on one vertical line). In that extreme case ATA is positive semidefinite. We have for the matrix ATA: If the center of mass is at the origin (i.e. C ti = C bi = 0) then ehen we solve the resulting system [ T c"t: ] [ ; ] = [ = : t i ] we get The fact that C = 0 indicates that the least squares line b = D t passes through the origin. For all points t o lie on the same line we must have one of the following 1. (from geometric considerations) b2 - bl - b3 - b2 t2 - tl...
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f02q1sol - MIT OpenCourseWare http://ocw.mit.edu 18.085...

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