01aut-fsols

# 01aut-fsols - MATH 51 FINAL EXAM SOLUTIONS (AUTUMN 2001) 1....

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MATH 51 FINAL EXAM SOLUTIONS (AUTUMN 2001) 1. Compute the following. (a) 1 2 1 2 1 0 1 0 0 - 1 Solution. 0 0 1 0 1 - 2 1 - 2 3 (b) The angle between - 1 4 1 and 2 - 2 1 . Solution. cos θ = v · w k v kk w k = - 9 3 18 = - 2 2 = θ = 3 π 4 (c) The area of the triangle with vertices (0 , 0 , 0), ( - 1 , 4 , 1) and (2 , - 2 , 1). Solution. The area of this triangle is half the area of the parallelogram gen- erated by v = - 1 4 1 and w = 2 - 2 1 . Since v × w = 6 3 - 6 , the area of the triangle is 1 2 k v × w k = 9 2 . Equivalently, using the result from part (b), the triangle has a base of k w k = 3 and a height of k v k sin θ = 3, so the area is 1 2 · 3 · 3 = 9 2 . 2. Let A = 1 2 1 2 1 3 2 4 7 18 11 22 . (a) For which vectors b = b 1 b 2 b 3 does the equation A x = b have a solution? Express your answer as one or more equations of the form ? b 1 +? b 2 +? b 3 =?. 1

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Solution. Reducing the augmented matrix for the system A x = b gives 1 2 1 2 1 3 2 4 7 18 11 22 b 1 b 2 b 3 -→ 1 2 1 2 0 1 1 2 0 4 4 8 b 1 b 2 - b 1 b 3 - 7 b 1 -→ 1 2 1 2 0 1 1 2 0 0 0 0 b 1 b 2 - b 1 b 3 - 7 b 1 - 4( b 2 - b 1 ) The system is therefore consistent (i.e. b is in C ( A )) if and only if - 3 b 1 - 4 b 2 + b 3 = 0. (b) Find a basis for the null space of A . Solution. Continuing with the elimination from part (a) gives rref( A ) = 1 0 - 1 - 2 0 1 1 2 0 0 0 0 . so a basis for N ( A ) is 1 - 1 1 0 , 2 - 2 0 1 . (c) Find a basis for the column space of A . Solution. Since the pivots of rref( A ) are in the ﬁrst two columns, the ﬁrst two columns of A 1 1 7 , 2 3 18 form a basis for C ( A ). (d) What is the rank of A ? Solution. 2 3. (a) Let b = 1 2 3 4 5 v 1 = 1 3 0 1 2 v 2 = 3 5 2 1 4 v 3 = 1 0 4 3 4 . 2
Express b as a linear combination of v 1 , v 2 and v 3 . Solution. Since rref 1 3 1 1 3 5 0 2 0 2 4 3 1 1 3 4 2 4 4 5 = 1 0 0 3 2 0 1 0 - 1 2 0 0 1 1 0 0 0 0 0 0 0 0 , it follows that b = 3 2 v 1 - 1 2 v 2 + v 3 . (b) Assume A 1 2 3 4 = 2 0 - 1 and rref( A ) = 1 0 0 5 0 0 1 - 7 0 0 0 0 . Find all solutions of A x = 2 0 - 1 . Solution.

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## This note was uploaded on 01/12/2010 for the course MATH 51 at Stanford.

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01aut-fsols - MATH 51 FINAL EXAM SOLUTIONS (AUTUMN 2001) 1....

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