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reviewfinal - Linear Algebra Review Problems for Final Exam...

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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 Gramm-Schmidt process to the set { 1 1 1 , - 1 1 0 , - 1 0 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
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mial that fits the points ( - 2 , - 4) , ( - 1 , - 1) , (0 , 0) , (1 , 0) , (2 , 0) 8. (12 points) Let T : P 3 P 2 be the linear transformation T ( f ( x )) = f ( x ).
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