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final_1806_s05

# final_1806_s05 - 18.06 Professor Strang Final Exam Grading...

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1 2 3 4 5 6 7 8 9 10 18.06 Professor Strang Final Exam May 16, 2005 Grading Your PRINTED name is: Closed book / 10 wonderful problems. Thank you for taking this course ! There is an extra blank page at the end.

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1 (10 pts.) Suppose P 1 , . . . , P n are points in R n . The coordinates of P i are ( a i 1 , a i 2 , . . . , a in ). We want to find a hyperplane c 1 x 1 + · · · + c n x n = 1 that contains all n points P i . (a) What system of equations would you solve to find the c ’s for that hyperplane ? (b) Give an example in R 3 where no such hyperplane exists (of this form), and an example which allows more than one hyperplane of this form. (c) Under what conditions on the points or their coordinates is there not a unique interpolating hyperplane with this equation ? 2
2 (10 pts.) (a) Find a complete set of “special solutions” to Ax = 0 by noticing the pivot variables and free variables (those have values 1 or 0). 1 2 3 4 5 A = 1 2 3 4 6 . 0 0 0 0 0 (b) and (c) Prove that those special solutions are a basis for the nullspace N ( A ).

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