MA 405 Final Examination Solutions

MA 405 Final - Lin Alg Matrices Final Examination Spring 1996 Your Name SOLUTION For purpose of anonymous grading please do not write your name on

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Lin Alg & Matrices Final Examination Spring 1996 Your Name: SOLUTION For purpose of anonymous grading, please do not write your name on the subsequent pages. This examination consists of 5 questions, each question counting for the given number of points, adding to a total of 24 points. Please write your answers in the spaces indicated, or below the questions (using the back of the sheets if necessary). You are allowed to consult two 8 . 5 0 × 11 0 sheets with notes, but not your book or your class notes. If you get stuck on a problem, it may be advisable to go to another problem and come back to that one later. You will have 2 hours to do this test. Good luck! Problem 1 2 3 4 5 Total 1
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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 = ( A ) = NS ( A T ). B. The eigenspace is ( 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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This note was uploaded on 06/11/2010 for the course MA 405 taught by Professor Staff during the Spring '08 term at N.C. State.

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MA 405 Final - Lin Alg Matrices Final Examination Spring 1996 Your Name SOLUTION For purpose of anonymous grading please do not write your name on

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