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MA 405 Final Examination Solutions

# MA 405 Final Examination Solutions - Lin Alg Matrices Final...

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Problem 1 (7 points): Please answer the following questions and justify your answers briefly. (a, 2pts) Consider the following 2 problems: A. Given a basis for a subspace of R n find a basis for its orthogonal complement. B. Given a matrix and one of its eigenvalues, find a basis for the corresponding eigenspace. Both problems can be reduced to the problem of finding a basis for the nullspace of a matrix. Please explain what those nullspace problems look like. A. Let v 1 , . . . , v r be a basis for V R n . Write A = [ v 1 | v 2 | . . . | v r ]. Then V = CS ( A ), hence V = CS ( A ) = NS ( A T ). B. The eigenspace is NS ( A - λ 1 I ) where λ 1 is the eigenvalue. (b, 1pts) For the following weighted inner product on R 3 , h x , y i = x 1 y 1 + 2 x 2 y 2 + x 3 y 3
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