final_1806_s05

# Final_1806_s05 - 18.06 Professor Strang Final Exam Grading 1 Your PRINTED name is Closed book 10 wonderful problems Thank you for taking this

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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( 1 0 p t s . ) 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( 1 0 p t s . ) (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). 12345 A = 12346 . 00000 (b) and (c) Prove that those special solutions are a basis for the nullspace N ( A ).

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## This note was uploaded on 01/14/2011 for the course EECS 18.06 taught by Professor Strang during the Spring '05 term at University of Michigan.

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Final_1806_s05 - 18.06 Professor Strang Final Exam Grading 1 Your PRINTED name is Closed book 10 wonderful problems Thank you for taking this

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