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e608fk - MAS 3105 Quiz 6 Nov 21, 2008 Prof. S. Hudson 1)...

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MAS 3105 Nov 21, 2008 Quiz 6 Prof. S. Hudson 1) Let V = C [0 , 2 π ] with inner product h f,g i = 1 π R 2 π 0 f ( x ) g ( x ) dx . a) Show that 1 is orthogonal to cos(5 x ) in V . b) Compute || sin(3 x ) || . [If you have forgotten some Calculus/Trig skills, you can replace sin(3 x ) by x 2 , for partial credit]. 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) Derive the formula for the projection matrix P for a Least Squares problem (it represents projection onto R ( A )). Then, show that P 2 = P . c) Show that if rank ( A ) = n (= number of columns), then the normal equations have a unique solution ˆ x . (You may use the textbook proof, or repeat HW 5.2.13, etc) Remarks and Answers: The average was about 40/60. The scale for this quiz is: A-’s
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