11test1a

# 2 consider the linear system w x y z 1 w x y z 1 2 w

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2. Consider the linear system w + x + y + z = 1 w - x + y + z = 1 2 w - 3 x + 2 y + 2 z = 2 . a ) What is the rank of the matrix of coeﬃcients? Answer: The rank is 2. This is clear once we row-reduce the augmented matrix: 1 1 1 1 1 1 - 1 1 1 1 2 - 3 2 2 2 (2) - (1) , (3) - 2(1) ----------→ 1 1 1 1 1 0 - 2 0 0 0 0 - 5 0 0 0 (3) - 5 / 2(2) -------→ 1 1 1 1 1 0 - 2 0 0 0 0 0 0 0 0 b ) Does this system have any solutions? Answer: Yes. c ) If the system has one or more solutions, describe them. Is the solution unique? Answer: Any vector of the form (1 - y - z, 0 ,y,z ) T will satiﬁsy the system. The solution is not unique.

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MATHEMATICAL ECONOMICS EXAM #1, SEPTEMBER 26, 2011 Page 2 3. Consider the set T = { x R 3 : k x k = 1 } ⊂ R 3 . a ) Is the set T open, closed, or neither when considered as a subset of R 3 ? Answer: The set is closed. b ) Prove your answer in part (a). Answer: One way to prove this is to note that f ( x ) = k x k is continuous, so if x n x , then k x n k → k vx k . Since k x n k = 1, k x k = 1 and x T .
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2 Consider the linear system w x y z 1 w x y z 1 2 w 3 x 2...

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