01aut-m1sols

# 01aut-m1sols - MATH 51 MIDTERM 1 SOLUTIONS(AUTUMN 2001 2 1...

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MATH 51 MIDTERM 1 SOLUTIONS (AUTUMN 2001) 1. Suppose u = 2 5 1 , v = - 1 0 1 , θ is the angle between them, and A = 1 2 3 4 5 6 . Compute: (a) (2 points) 2 u - 3 v , Solution. 2 u - 3 v = 7 10 - 1 (b) (2 points) k u kk v k cos θ , Solution. k u kk v k cos θ = u · v = - 1 (c) (2 points) k u kk v k sin θ , Solution. Since u × v = 5 - 3 5 , k u kk v k sin θ = k u × v k = 59. (d) (2 points) k u k , Solution. k u k = 30. (e) (2 points) A u . Solution. A u = 15 39 1

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true or false . No explanation is required. You should be able to answer all of these questions without doing any calculations. (a) 1 2 3 4 , 2 3 . 4 9 10 , π 14 - 1 1 . 2 , 1 . 3 - 4 3 - 94 , 5 9 10 - 764 is a linearly independent set. Solution. False. (Any set of more than 4 vectors in R 4 is linearly dependent.) (b) π e 7 / 3 4 , - 2 π - 2 e - 14 / 3 - 8 , 1 2 3 0 is a linearly independent set. Solution. False. (The second vector is - 2 times the ﬁrst.) (c) π e 7 / 3 4 , 11 3 2 5 , 1 2 3 0 spans R 4 . Solution. False. (At least four vectors are needed to span R 4 .) (d) 1 0 0 , 3 7 8 , 0 1 0 , 0 0 1 , 9 8 2 spans R 3 . Solution. True. (The ﬁrst, third and fourth vectors alone clearly span R 3 .) (e) If A is an m × n matrix and b R m is an arbitrary vector, then the set of solutions to A x = b is always a linear subspace of R n . Solution.
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01aut-m1sols - MATH 51 MIDTERM 1 SOLUTIONS(AUTUMN 2001 2 1...

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