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e607k - 1 The normal equations are A T A ˆ x = A T b so...

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MAS 3105 April 5, 2007 Quiz 6 and Key Prof. S. Hudson 1) Write out explicitly the normal equations for this inconsistent system. But you don’t have to solve them, to actually find the least squares solution. - x 1 + x 2 = 10 2 x 1 + x 2 = 5 x 1 - 2 x 2 = 20 2) Let V = C [0 , 1], with inner product f, g = 1 0 f ( x ) g ( x ) dx . Use this to find || x + 1 || . 3) Choose ONE of these. a) [HW 5.2.13a,b] Assume x N ( A T A ). Prove that A x N ( A T ) and A x R ( A ) and x N ( A ). b) State and prove the Fundamental Subspace Theorem (5.2.1). Remarks and Answers: The average was about 40/60. The unofficial scale is A’s 45-60, with 6 points per letter below that. I also computed your semester average, based on your best 5 out 6 quiz scores so far, and wrote a corresponding letter grade in the upper corner of your quiz. The scale for that was: A’s 240-300, B’s 210-240, etc (30 points per letter). I will include HW and MHW later, of course. If you have not handed-in MHW, for example, your actual grade is probably worse than what I wrote.
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Unformatted text preview: 1) The normal equations are A T A ˆ x = A T b , so you substitute into that, and get ± 6-1-1 6 ² ˆ x = ± 20-25 ² I intended for you to multiply out A T A and A T b , but gave full credit without that. I gave 2 points extra credit if you did so. 2) In R n , we compute the norm of a vector from || x || = ( x T x ) 1 / 2 . In an inner product space, we use h x,x i instead. ( R 1 ( x + 1) 2 dx ) 1 / 2 = p 7 / 3. 3a) [This was HW, and we also did it in class]. The part about Ax ∈ R ( A ) is from basic deﬁnitions (see pages 36-37). And x ∈ N ( A T A ) means A T A x = which means A x ∈ N ( A T ). By the main thm (5.2.1), these show that A x ∈ S ∩ S ⊥ = { } (see Remark 1 on page 227). 3b) See the text. You should really include the paragraph above Theorem 5.2.1, too, though I wasn’t very strict about that this time. 1...
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