b With x x 1 x 2 x 3 this is the same as Ax bfor some x R 3 To see whether the

B with x x 1 x 2 x 3 this is the same as ax bfor some

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b With ~x = x 1 x 2 x 3 , this is the same as A~x = ~ b for some ~x R 3 . To see whether the linear system A~x = ~ b has a solution we row-reduce the augmented matrix 1 0 5 2 - 2 1 - 6 - 1 0 2 8 6 7→ 1 0 5 2 0 1 4 3 0 0 0 0 The equivalent system has (infinitely many) solutions, x 1 = 2 - 5 x 3 , x 2 = 3 - 4 x 3 , x 3 R is free For example x 1 = 2, x 2 = 3, x 3 = 0 is one such solution, and it is easy to check that it indeed gives the right answer: 2 1 - 2 0 + 3 0 1 2 + 0 5 - 6 8 = 2 - 1 6 (1) So ~ b is indeed in the span of the columns of A , and one such linear combination of the columns is written in equation (1). Answer: ~ b is in the span of the columns of A 2. Matrix A = 1 2 3 4 5 2 2 4 6 1 3 2 5 8 2 4 - 3 1 5 3 is row equivalent to matrix B = 1 0 1 2 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 . Use this fact to describe all solutions of equation A~x = ~ 0 in parametric vector form (or as a span). Answer: This problems is similar to problems 8, 11 in Section 1.5. Equation A~x = 0 w has the same solution set as equation B~x = 0, and matrix B is in reduced echelon form. The free variables are x 3 , x 4 and the basic variables
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