MIT18_06S10_Final_Exam

# MIT18_06S10_Final_Exam - 18.06 Final Exam Professor Strang...

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Unformatted text preview: 18.06 Final Exam May 18, 2010 Professor Strang Your PRINTED name is: 1. Your recitation number is 2. 3. 4. 5. 6. 7. 8. 9. 1. (12 points) This question is about the matrix ⎡ 1 2 1 A = 2 4 1 4 ⎣ . 3 6 3 9 (a) Find a lower triangular L and an upper triangular U so that A = LU. (b) Find the reduced row echelon form R = rref ( A ) . How many independent columns in A ? (c) Find a basis for the nullspace of A . (d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 2. (11 points) This problem finds the curve y = C + D 2 t which gives the best least squares fit to the points ( t, y ) = (0 , 6) , (1 , 4) , (2 , 0) . (a) Write down the 3 equations that would be satisfied if the curve went through all 3 points. (b) Find the coeﬃcients C and D of the best curve y = C + D 2 t . (c) What values should y have at times t = 0 , 1 , 2 so that the best curve is y = 0? 3. (11 points) Suppose Av i = b i for the vectors v 1 , . . . , v n and b 1 , . . . , b n in R n . Put the v ’s into the columns of V and put the b ’s into the columns of B....
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MIT18_06S10_Final_Exam - 18.06 Final Exam Professor Strang...

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