e606k - in MHW, or have very unusual HW/MHW scores, this...

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MAS 3105 June 15, 2006 Quiz 6 and Key Prof. S. Hudson 1) Let S be the subspace of R 3 spanned by x = (2 , 1 , 1) T . Find a basis of S . 2) Find the distance from the point P(1,2,3) to the plane x + y + 2 z = 0. 3) Choose ONE of these. a) [based on MHW 4.1] Suppose that F = { w 1 , w 2 , w 3 } = { e 3 , e 1 + e 2 , e 1 } is a basis for R 3 , and L : R 3 R 3 . Suppose L ( w 1 ) = 4 w 1 and L ( w 2 ) = 4 w 1 + 3 w 2 and L ( w 3 ) = 4 w 1 + 3 w 2 + 2 w 3 . Find the matrix representation of L with respect to F . b) If A and B are similar, then rank A = rank B . c) State and prove (explain) the normal equations used in Least Squares problems. Remarks and Answers: The average was about 48/60, very good for a Quiz 6. In the upper right corner, I averaged your best 5 quiz grades and gave you an approx semester letter grade. This does not yet include your HW or MHW grades. If you have not handed
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Unformatted text preview: in MHW, or have very unusual HW/MHW scores, this estimate is probably not very accurate. 1) { [-1 / 2 1 0] T , [-1 / 2 0 1] T } 2) The scalar projection of P onto the normal vector N = 9 / √ 6. 3a) As in the HW/MHW, you can ignore the e j part. The first column of the answer is a 1 = [4 0 0] T , etc. 3b) From A = S-1 BS , get SA = BS . From HW, we know N ( SA ) = N ( A ) so these two have the same nullity, the same width, thus the same rank. Now, show rank ( BS )= rank ( B ). Since the S is on the right, it is not true that N ( BS ) = N ( B ). So, you need a little trick (take the transpose first, which doesn’t change the rank). 1...
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This note was uploaded on 12/26/2011 for the course MAS 3105 taught by Professor Julianedwards during the Spring '09 term at FIU.

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