e605k - a) Prove thm 5.1.1; that for nonzero vectors x , y...

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MAS 3105 June 16, 2005 Quiz 6 Prof. S. Hudson 1) Answer True or False ( A is an m x n matrix, and S,T are subspaces of R n ): If rank A = n , then A T A is nonsingular. If S T = { 0 } then S T = R n . N ( A T ) R ( A ) = R m . If ˆ x is a least squares solution to Ax = b, then A ˆ x is the projection of b onto R(A). Every norm on P 3 satisfies the triangle inequality. 2) Let L : P 3 P 3 be the transformation L ( p ( x )) = xp 0 ( x ) + p (0). Find the matrix representation of L with respect to the basis [1 ,x,x 2 ]. 3) Choose ONE of these to prove.
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Unformatted text preview: a) Prove thm 5.1.1; that for nonzero vectors x , y R 2 , x T y = || x || || y || cos( ). b) State and prove the normal equations, used to solve least squares problems. You can start from a picture and can assume that b-p R ( A ) . Answers: 1) TFTTT 2) A = 1 0 0 0 1 0 0 0 2 3) Both are in the book. 1...
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