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# hw5 - Solutions of Selected Problems in HW5 3.2.10(b(1 0...

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Solutions of Selected Problems in HW5 3.2.10 (b) { (1 , 0 , 0) , (0 , 1 , 1) , (1 , 0 , 1) , (1 , 2 , 3) } We have to check existence of C 1 , · · · , C 4 such that C 1 1 0 0 + C 2 0 1 1 + C 3 1 0 1 + C 4 1 2 3 = x 1 x 2 x 3 for arbitrary x 1 , x 2 , x 3 . This gives the augmented matrix 1 0 1 1 x 1 0 1 0 2 x 2 0 1 1 3 x 3 and its RREF is 1 0 0 0 x 1 + x 2 - x 3 0 1 0 2 x 2 0 0 1 1 - x 2 + x 3 . This system is consistent and always has a solution no matter what x 1 , x 2 and x 3 . Thus it spans R 3 . (c) { (2 , 1 , - 2) , (3 , 2 , - 2) , (2 , 2 , 0) } The corresponding augmented matrix is 2 3 2 x 1 1 2 2 x 2 - 2 - 2 0 x 3 and its RREF is 1 0 - 2 2 x 1 - 3 x 2 0 1 2 - x 1 + 2 x 2 0 0 0 2 x 1 - 2 x 2 + x 3 . The system is inconsistent if 2 x 1 - 2 x 2 + x 3 6 = 0. This implies that no linear combination of the above three vectors can make a vector ( x 1 , x 2 , x 3 ) with 2 x 1 - 2 x 2 + x 3 6 = 0.

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hw5 - Solutions of Selected Problems in HW5 3.2.10(b(1 0...

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