Let V denote the set of all solutions to the system of linear equations x1−x2 + 2x4−3x5+ x6= 0 2x1−x2−x3 + 3x4−4x5 + 4x6 = 0. (a) Show that S = {(0,−1, 0, 1, 1, 0), (1, 0, 1, 1, 1, 0)} is a linearly independent subset of V. (b) Extend S to a basis for V.

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Let V denote the set of all solutions to the system of linear equations

x1−x2 + 2x4−3x5+ x6= 0

2x1−x2−x3 + 3x4−4x5 + 4x6 = 0.

(a) Show that S = {(0,−1, 0, 1, 1, 0), (1, 0, 1, 1, 1, 0)} is a linearly independent subset of V.

(b) Extend S to a basis for V.

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