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Unformatted text preview: Linear Algebra Review Problems for Final Exam Note: The final will consist of 100 points with roughly equivalent point as signments. 1. (20 points) Given A = 1 0 1 0 1 0 1 1 1 (a) Compute the characteristic polynomial of A . (b) Find the eigenvalues of A . (c) Find a basis for R 3 consisting of eigenvectors of A . (d) Find an invertible matrix P and a diagonal matrix D such that P 1 AP = D . (e) Compute P 1 . (f) Write A as a product of P , P 1 , and D (in the correct order!) and use this information to compute A 6 . 2. (8 points) Explain why the matrix A = 0 1 0 0 0 1 0 0 0 cannot be diagonalized. 3. (5 points) Suppose a square matrix A satisfies A 3 = 2 A 2 A . What are the possible eigenvalues of A ? 4. (10 points) Suppose A is a 5 × 3 matrix with orthonormal columns. Eval uate (a) det( A T A ). (b) det( AA T ). (c) det( A ( A T A ) 1 A T ). 5. (12 points) Apply the GrammSchmidt process to the set { 1 1 1 ,  1 1 ,  1 1 } . 6. (8 points) Explain why the matrices A = 1 1 1 1 1 1 1 1 1 and B = 1 1 1 1 2 2 1 2 2 are NOT similar. 7. (12 points) Find the best fit (in least squares sense) quadratic polyno 1 mial that fits the points ( 2 , 4) , ( 1 , 1) , (0 , 0)...
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This note was uploaded on 10/18/2011 for the course S 2010D taught by Professor Peters during the Spring '10 term at Columbia.
 Spring '10
 Peters

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