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ef08fk - MAS 3105 Final Exam and key Prof S Hudson 1(10pts...

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MAS 3105 Dec 10, 2008 Final Exam and key Prof. S. Hudson 1) (10pts) Suppose A is nxn, real, and skew-symmetric (so, A T = - A ). Show that: a) A is normal. b) If n is odd, A must be singular. 2) (10 pts) a) Find the least squares solution to A x = b , using this QR factorization: A = 1 - 2 1 2 0 1 2 - 4 2 4 0 0 = 1 5 1 - 2 - 4 2 1 2 2 - 4 2 4 2 - 1 5 - 2 1 0 4 - 1 0 0 2 where b = - 1 1 1 - 2 2b) Let S = R ( A ) and use part a) to find proj S b . 3) (10pts) Choose ONE (a fairly short proof) a) State and prove the Spectral Theorem b) If Y V is a subspace, then so is Y . c) If A and B are nonsingular, then so is AB (state and prove a formula for its inverse). 4) (10pts) Factor A into XDX - 1 where D is diagonal (compute the 3 factors - and I’d suggest checking them by multiplication). A = 2 - 8 1 - 4 5) (10pts) Let S R 3 x 3 be the vector space of all symmetric 3x3 matrices. Find dim(S) and explain briefly. 6) (5pts) Suppose A is 2x2 and tr(A)= 3 and det(A) = 2. Find the 2 real eigenvalues of A .
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