Final Exam Solutions - Final Exam December Final Exam...

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Final Exam, December , Final Exam Linear Algebra, Dave Bayer, December , [ 1 ] Find the intersection of the following two a ne subspaces of R 3 . x y z = 1 1 1 + 1 0 1 1 0 1 a b x y z = 1 2 2 + 1 0 0 1 0 1 c d
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Final Exam, December , [ 2 ] Find an orthogonal basis for the subspace of R 4 de ned by the equation w + x - y - z = 0 . Extend this basis to a orthogonal basis for R 4 .
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Final Exam, December , [ 3 ] Find the determinant of the matrix 2 1 0 0 0 0 0 2 2 1 0 0 0 0 0 2 2 1 0 0 0 0 0 2 2 1 0 0 0 0 0 2 2 1 0 0 0 0 0 2 2 1 0 0 0 0 0 2 2
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Final Exam, December , [ 4 ] Solve the di erential equation y 0 = Ay where A = 2 1 3 0 , y ( 0 ) = 1 0 λ = - 1, 3 e At = e - t 4 1 - 1 - 3 3 + e 3 t 4 3 1 3 1 y = e - t 4 1 - 3 + e 3 t 4 3 3
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Final Exam, December , [ 5 ] Express the quadratic form - 4 xy + 3 y 2 as a sum of squares of othogonal linear forms. λ = - 1, 4 A = 0 - 2 - 2 3 = - 1 5 4 2 2 1 + 4 5 1 - 2 - 2 4 - 4 xy + 3 y 2 = x y 0 - 2 - 2 3 x y = - 1 5 ( 2 x + y ) 2 + 4 5 ( x - 2 y ) 2
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Final Exam, December , [ 6 ] Solve the recurrence relation f ( 0 ) = a , f ( 1 ) = b , f ( n ) = 3 f ( n - 1 ) - 2 f ( n - 2 ) f ( n + 1 ) f ( n ) = 3 - 2 1 0 n b a = - 1 2 - 1 2 b a + 2 n 2 - 2 1 - 1 b a f ( n ) = ( - b + 2 a ) + 2 n ( b - a )
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Final Exam, December
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