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gauss_examples

# gauss_examples - Example II.3(Feldman s notes 2x1 x2 3x3 =...

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Example II.3 (Feldman 0 s notes) 2 x 1 + x 2 + 3 x 3 = 1 4 x 1 +5 x 2 + 7 x 3 = 7 2 x 1 - 5 x 2 + 5 x 3 = - 7 2 1 3 4 5 7 2 - 5 5 1 7 - 7 7 23 - 5 (1) (2) - 2(1) (3) - (1) 2 1 3 0 3 1 0 - 6 2 1 5 - 8 7 9 - 12 (1) (2) (3) + 2(2) 2 1 3 0 3 1 0 0 4 1 5 2 7 9 6 (3) = 4 x 3 = 2 = x 3 = 1 2 (2) = 3 x 2 + 1 2 = 5 = x 2 = 3 2 (1) = 2 x 1 + 3 2 + 3 × 1 2 = 1 = x 1 = - 1 Check 2( - 1) + 3 2 + 3 × 1 2 = 1 4( - 1) + 5 × 3 2 + 7 × 1 2 = 7 2( - 1) - 5 × 3 2 + 5 × 1 2 = - 7

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Example II.4 (Feldman 0 s notes) x 1 + 2 x 2 + x 3 + 2 x 4 + x 5 = 1 2 x 1 + 4 x 2 + 4 x 3 + 6 x 4 + x 5 = 2 3 x 1 + 6 x 2 + x 3 + 4 x 4 + 5 x 5 = 4 x 1 + 2 x 2 + 3 x 3 + 5 x 4 + x 5 = 4 1 2 1 2 1 2 4 4 6 1 3 6 1 4 5 1 2 3 5 1 1 2 4 4 8 19 23 16 (1) (2) - 2(1) (3) - 3(1) (4) - (1) 1 2 1 2 1 0 0 2 2 - 1 0 0 - 2 - 2 2 0 0 2 3 0 1 0 1 3 8 3 - 1 8 (1) (2) (3) + (2) (4) - (2) 1 2 1 2 1 0 0 2 2 - 1 0 0 0 0 1 0 0 0 1 1 1 0 1 3 8 3 2 5 (1) (2) (4) (3) 1 2 1 2 1 0 0 2 2 - 1 0 0 0 1 1 0 0 0 0 1 1 0 3 1 8 3 5 2
Back Solution - the direct method (4) x 5 = 1 (3) x 4 + 1 = 3 x 4 = 2 (2) 2 x 3 + 2 × 2 - 1 = 0 x 3 = - 3 2 (1) x 1 + 2 x 2 - 3 2 + 2 × 2 + 1 = 1 x 1 + 2 x 2 = - 5 2 x 2 = t , arbitrary x 1 = - 5 2 - 2 t Back Solution - the row reduction method (1) - (4) (2) + (4) (3) - (4) (4) 1 2 1 2 0 0 0 2 2 0 0 0 0 1 0 0 0 0 0 1 0 1 2 1 6 5 3 2 (1) - 2(3) (2) - 2(3) (3) (4) 1 2 1 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 1 - 4 - 3 2 1 0 - 1 3 2 (1) (2) / 2 (3) (4) 1 2 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 - 4 - 3 2 2 1 0 - 1 2 3 2 (1) - (2) (2) (3) (4) 1 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 - 5 / 2 - 3 / 2 2 1 x 2 = t, x 1 = - 5 2 - 2 t x 3 = - 3 2 x 4 = 2 x 5 = 1

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Example II.5 (Feldman 0 s notes) – Check ( - 5 2 - 2 t ) + 2
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