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04aut-f - MATH 51 FINAL EXAM 1 Consider the matrices 0011 A...

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MATH 51 FINAL EXAM (December 6, 2004) 1. Consider the matrices A = 0 0 1 1 1 3 1 2 2 6 1 3 and R = 1 3 0 1 0 0 1 1 0 0 0 0 . The matrix R is the row reduced echelon form of A . (You do not need to check this.) 1(a). Find a basis for the column space of A . 1(b). Find a basis for the null space of R . 1(c). Note that A 1 1 1 1 = 2 7 12 . Find all solutions to A x = 2 7 12 . 2. Consider the following system of equations: x 2 + x 3 = a x 1 + x 2 + 2 x 3 = b x 1 + 2 x 2 + 3 x 3 = c 2 x 1 + 3 x 2 + 5 x 3 = d Find the condition(s) on a , b , c , and d , for the system to have a solution. (Your answer should be one or more equations of the form ? a +? b +? c +? d =?.) 3(a). Find all eigenvalues of the matrix A = 1 0 2 7 3 5 2 0 1 . 3(b). Consider the matrix M = 3 1 1 0 4 1 0 0 3 . Note that v 1 = 1 1 0 is an eigenvec- tor with eigenvalue 4. Find eigenvectors v 2 and v 3 so that v 1 , v 2 , and v 3 form a basis for R 3 .
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