oldfinal - A x = b , where A = 1 2 1-1 1 1 1 and b = 1 1 1...

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Math 290 - Final exam, Dec. 14, 2006 Name: Directions: Please be careful and check your work. As usual, no calculators are permitted, and you should show all your work on these pages. If you need more space, use the back sides of the paper 1. (25 pts) Definitions: linearly dependent rank of a matrix basis of a vector space eigenvalue null space 2. (20 pts) Write ± 5 4 1 1 ² as the product of elementary matrices. Don’t multiply the matrices out. 1
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3. (20 pts) Evaluate the determinant of 3 0 1 2 - 1 1 1 2 0 , by using row operations. 4. (20 pts) Show that 1 2 3 is not in the span of 1 0 1 , - 1 1 1 . 2
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5. (20 pts) Show that ±² 1 1 ³ , ² 1 - 1 ³´ form a basis for R 2 . 6. (20 pts) If A = 1 - 1 1 2 1 0 1 - 4 3 , then Find rank( A ). Find N ( A ). Find a basis for the range of A . 3
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7. (20 pts) Find an orthonormal basis for the null space of A = ( 1 1 1 ) . 8. (20 pts) Find the least squares solution to
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Unformatted text preview: A x = b , where A = 1 2 1-1 1 1 1 and b = 1 1 1 1 .. (Use the normal equation.) 4 9. (20 pts) Find the eigenvalues and eigenvectors of the matrix 2-3-3 2 . Then nd a matrix P so that P-1 AP = 5-1 . 10. (15 pts) Answer the following questions. Be concise, but give the whole answer. If A is m n, rank( A ) = n, and A x = b has a solution, then the solution is unique. Why? If A x = b is inconsistent, then we can get a least squares approximate solution by solving A x = b r . How is b r obtained from b ? If A is a square matrix and can be row reduced to the identity, then ( A | I ) is reduced by the same operations to ( I | B ). How do we know that B = A-1 ? 5...
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oldfinal - A x = b , where A = 1 2 1-1 1 1 1 and b = 1 1 1...

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