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Unformatted text preview: (d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. Answer: 1 2 3 1 1 2 x = + x 2 + x 4 1 1 1 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. Answer: C + 1D = 6 C + 2D = 4 C + 4D = (b) Find the coecients C and D of the best curve y = C + D 2 t . Answer: 1 1 1 1 1 3 7 A T A = 1 2 = 1 2 4 7 21 1 4 6 1 1 1 10 A T b = 4 = 1 2 4 14 Solve A T Ax = A T b : 3 7 C 10 C 1 21 10 8 = gives = 7 = . 7 21 D 14 D 14 7 3 14 2 (c) What values should y have at times t = 0 , 1 , 2 so that the best curve is y = 0? Answer: The projection is p = (0 , , 0) if A T b = 0 . In this case, b = values of y = c (2 , 3 , 1) . 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. (a) Write those equations Av i = b i in matrix form. What condition on which vectors allows A to be determined uniquely? Assuming this condition, find A from V and B. Answer: A [ v 1 v n ] = [ b 1 b n ] or AV = B. Then A = BV 1 if the v s are independent. (b) Describe the column space of that matrix A in terms of the given vectors. Answer: The column space of A consists of all linear combinations of b 1 , , b n . (c) What additional condition on which vectors makes A an invertible matrix? Assuming this, find A 1 from V and B ....
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 Fall '10
 Strang
 Linear Algebra, Algebra, Least Squares

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