e603k - Instead of using Ch.1, some people guessed, but...

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MAS 3105 2003 Quiz 6 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) Let L be the derivative operator on P 3 . So, L ( p ( x )) = p 0 ( x ). Find the matrix represen- tation of L with respect to the basis [1 ,x,x 2 ]. 3) Choose ONE of these. a) Prove thm 5.1.1; that for nonzero vectors x , y R 2 , x T y = || x || || y || cos( θ ). b) If A and B are similar, then A 3 and B 3 are similar. c) Derive the normal equations, used to solve least squares problems. Answers: 1) Compute N ([2 1 1]) using Ch.1 methods (set x 3 = α , etc) and get [ - 1 / 2 1 0] T and [ - 1 / 2 0 1] T . Since a basis is not unique, other answers are possible.
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Unformatted text preview: Instead of using Ch.1, some people guessed, but this didnt usually work out. Re-member that dim S + dim S = 3, so the answer should be a list of two vectors. 2) Since L : P 3 P 3 , and dim P 3 = 3, the matrix should be 3x3. Compute L (1) = 0 = [0 0 0] T which is column 1, etc. Get A = 1 2 3) These are in the text, except that 3b) was HW: Assume A = S-1 BS . Cube both sides and cancel the SS-1 s and get A 3 = S-1 B 3 S . So, A 3 is similar to B 3 . 1...
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