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solset2 - Problem Set#2 Solutions 2.4#33 If the three...

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Problem Set #2 Solutions § 2.4 #33: If the three solutions in Question 32 are ~ x 1 = (1 , 1 , 1) and ~ x 2 = (0 , 1 , 1) and ~ x 3 = (0 , 0 , 1) , solve A~x = ~ b when ~ b = (3 , 5 , 8) . Challenge problem: What is A ? Solution: Needed: A ~ x 1 = 1 0 0 , A ~ x 2 = 0 1 0 , A ~ x 3 = 0 0 1 If we let ~ x 1 = 1 1 1 , ~ x 2 = 0 1 1 , ~ x 3 = 0 0 1 And since we know A~x = b , we know the following: A~x = 3 5 8 = 3 1 0 0 + 5 0 1 0 + 8 0 0 1 = 3( A ~ x 1 ) + 5( A ~ x 2 ) + 8( A ~ x 3 ) = A (3 ~ x 1 ) + A (5 ~ x 2 ) + A (8 ~ x 3 ) ~x = 3 ~ x 1 + 5 ~ x 2 + 8 ~ x 3 . Recommended: A ~ x 3 = A 0 0 1 = Last column of A = 0 0 1 A = a 11 a 12 0 a 21 a 22 0 a 31 a 32 1 ~ x 2 = 0 1 1 : A ~ x 2 = A 0 1 0 + 0 0 1 = A 0 1 0 + 0 0 1 = 0 1 0 A 0 0 1 = a 12 a 22 a 23 = 0 1 - 1 A = a 11 0 0 a 21 1 0 a 31 - 1 1 Continued...
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