e604k - c) Prove that if A is similar to B , and B is...

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MAS 3105 3/29/04 Quiz 6 Key Prof. S. Hudson 1) Let x = [1 2] T , w 1 = [1 0] T and w 2 = [1 1] T . So, F = { w 1 , w 2 } is a basis of R 2 and W = [ w 1 w 2 ] is a nonsingular 2x2 matrix. Suppose L : R 2 R 2 is linear and L ( w 1 ) = w 2 and L ( w 2 ) = w 1 + 2 w 2 a) Find the matrix representation of L with respect to F . b) Find the coordinate vector y of x with respect to F . c) Find the coordinate vector of L ( x ) with respect to F . 2) Let P 1 = (1 , 1 , 1), P 2 = (2 , 4 , - 1) and P 3 = (0 , - 1 , 5). Find a nonzero vector N that is orthogonal to P 1 P 2 and P 1 P 3 3) Choose ONE of these to prove (on the back). a) 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 ) . b) State and prove the Fundamental Subspace Theorem (5.2.1).
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Unformatted text preview: c) Prove that if A is similar to B , and B is similar to C , then A is similar to C . Answers: The average was about 35/60, though the problems do not seem very hard. 1) This problem is based on Matlab exercise 1, page 220 (see also Example 3, page 202, etc). A = 0 1 1 2 , y = W-1 1 2 = -1 2 , L ( x ) = A y = 2 3 2) This is HW 5.2.5: P 1 P 2 = [1 3-2] T and P 1 P 3 = [-1-2 4] T . Get N = [8-2 1] (any nonzero scalar multiple of this is also OK) by computing the nullspace of A = 1 3-2-1-2 4 3) See text/lectures. 1...
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