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00aut-f - MATH 51 FINAL EXAM(AUTUMN 2000 1 Let 1 u1 = 1 3 1...

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MATH 51 FINAL EXAM (AUTUMN 2000) 1. Let u 1 = 1 - 1 3 u 2 = 1 1 - 2 u 3 = 3 - 1 4 (a) (6 points) Find the dimension of span( u 1 , u 2 , u 3 ). (b) (8 points) Find all vectors v which are simultaneously orthogonal (i.e. perpen- dicular) to all three vectors u 1 , u 2 and u 3 . 2. (10 points) Suppose B = ( x, y ) is a point on the circle of radius 1 centered at the origin. That is, x and y satisfy x 2 + y 2 = 1. Let A = ( - 1 , 0), C = (1 , 0) and assume y 6 = 0 (so that B is not equal to A or C ). A B C Use dot products to show that angle ABC is a right angle. 3. Suppose A is a 5 × 5 matrix with rref( A ) = 1 0 - 1 4 0 0 1 2 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 For each part below, give the answer when possible. Otherwise answer “not enough information”. (a) (2 points) Find a basis for N ( A ). (b) (2 points) Find dim( N ( A )). (c) (2 points) Find a basis for C ( A ). (d) (2 points) Find dim( C ( A )). (e) (2 points) Find the rank of A . (f) (2 points) Find a vector b R 5 such that A x = b has no solutions. (g) (2 points) Are there vectors b R 5 such that A x = b has exactly one solution? 1
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(h) (2 points) Find the eigenvalues of A . 4. Let A = 1 1 0 2 1 2 2 0 3 (a) (5 points) Compute det( A ).
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