FinalTestReviewSpring2011

B find the most general solution of a w for this

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Unformatted text preview: value of number c vector w = ￿ belongs 1 2 0 2 5 4 −2 c to the column space of A? (b) Find the most general solution of A￿ = w for this value of c. x￿ (c) Without further calculation, identify a basis for a row space of A (i.e., for C (AT )). ( pts) Solution: (a) Vector w belongs to C (A) precisely when system A￿ = w has a ￿ x ￿ solution. Thus, we need to solve it. We will do it by reducing augmented matrix of this system: 11 0 2 1 1 0 2 −￿2 r 1 0 −2 4 0 2 0 0 1 4 −4 × 1/2 1 2 −2 2 −2 → → 1 2 0 0 0 2 0 −￿1 r 1 2 −2 −￿2 r 0 0 5 4 −2 c −5￿1 r 0 −1 −2 c − 10 +￿2 r 00 0 c − 12 For this system to have the solution we need to have c − 12 = 0, that is, c = 12. (b) We already have done all reduction work for this problem in part (a). The free variable is x3 , since the third column is the only non-pivot column. From the first reduced equation x1 − 2x3 = 4, we get x1 = 4 + 2x3 . The second equation x2 + x3 =−2 2 gives x2 = − − 2x. 2 3 x1 4 −2...
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