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B to find an eigenvector corresponding to ? 10 we

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b ) To find an eigenvector corresponding to λ = 10, we solve ± - 1 - 2 - 2 - 4 ² v = ± 0 0 ² . One solution is v = (2 , - 1) T . To find an eigenvector corresponding to λ = 5, we solve ± 4 - 2 - 2 1 ² u = ± 0 0 ² . One solution is u = (1 , 2) T . c ) Although u · v = 0, which shows the eigenvectors are orthogonal, they do not have unit norm. We divide by their norms to get orthonormal eigenvectors: ± 2 5 - 1 5 ² , ± 1 5 2 5 ² .
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MATHEMATICAL ECONOMICS MIDTERM #2, NOVEMBER 12, 2002 Page 2 3. Robinson Crusoe has utility function u ( x,y ) = x 2 + y 2 . Crusoe has a production possi- bilities set given by { ( x,y ) : x 0 ,y 0 ,x + y 2 4 } . a ) Is the production possibility set a closed set? Is it a bounded set? b ) Show that Crusoe’s problem of maximizing utility over his production set has a solution. Answer: a ) The set is closed. There are many ways to see this, one is to realize that the functions f ( x,y ) = x , g ( x,y ) = y , and h ( x,y ) = x + y 2 are all continuous. Then f - 1 ([0 , )), g - 1 ([0 , )) and h - 1 (( -∞ ,
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b To find an eigenvector corresponding to 10 we solve 1...

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