04win-f - MATH 51 FINAL EXAM Professors Clingher, Munson,...

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MATH 51 FINAL EXAM Professors Clingher, Munson, and White March 15, 2004 1 Consider the matrices A = 1 1 0 0 2 2 2 0 1 6 0 1 - 1 1 3 - 1 - 2 1 1 - 1 and R = 1 0 1 0 1 0 1 - 1 0 1 0 0 0 1 2 0 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 column space of R . 1(c). Find a basis for the nullspace of A . 2. Find all solutions of x 1 + 2 x 2 + x 3 + x 4 = 7 x 1 + 2 x 2 + 2 x 3 - x 4 = 12 2 x 1 + 4 x 2 + 6 x 4 = 4 . 3(a). Find all eigenvalues of the matrix A = 5 0 0 1 2 1 1 1 2 . 3(b). The matrix M = 5 - 6 - 6 - 1 4 2 3 - 6 - 4 has λ = 2 as one of its eigenvalues. (You need not check this.) Let V be the eigenspace corresponding to this eigenvalue. (In other words, V consists of all eigenvectors with eigenvalue 2 together with the origin.) Find a basis for V . 4. The velocity of a certain spaceship at time t is given by v ( t ) = (3 t 2 ,e t - 1 , 6 t ). At time t = 1, its position is (0 , 0 , 7). (a) Find the speed at time t .
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04win-f - MATH 51 FINAL EXAM Professors Clingher, Munson,...

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