20f-mt1-soln

# 20f-mt1-soln - MATH 20F MIDTERM 1 You have 50 minutes for...

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Unformatted text preview: MATH. 20F MIDTERM 1 You have 50 minutes for this exam. Please write legibly and show all working. No calculators are allowed. Write your name and ID number. Name: ID Number: (1) Let A = 3 1 2 1 0 a 2 a . (i) (10 points) Find all a ’s such that the columns of A span R 3 . The columns of A span R 3 if and only if Ax = b is consistent for all b ∈ R 3 . Perform Gaussian elimination: A = 3 1 2 1 0 a 2 a −−−−−→ R 1 ↔ R 2 1 0 a 3 1 2 2 a R 3 → R 3- 2 · R 1 −−−−−−−−−→ R 2 → R 2- 3 · R 1 1 0 a 0 1 2 − 3 a a − 2 a −−−−−−−−−→ R 3 → R 3- a · R 2 1 0 a 0 1 2 − 3 a 0 0 3 a 2 − 4 a So the columns of A span R 3 if and only if 3 a 2 − 4 a = a · (3 a − 4) negationslash = 0 , i.e. a negationslash = 0 or 4 / 3. (ii) (5 points) Assume that a = 0 and b = (1 , , 0) t , find the general solution of Ax = b ....
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20f-mt1-soln - MATH 20F MIDTERM 1 You have 50 minutes for...

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