2130_Chapter7_Notes

Since x 0 is always a solution the only possibilities

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Unformatted text preview: here are no solutions. (b) 0 · · · 0 | 0 is equivalent to 0 = 0. (Two or more equations are redundant.) If row (b) occurs without row (a), there are infinitely many solutions. 4 / 13 Example 1 Equivalent to: x1 − x3 = 0 3x1 + x2 + x3 = 1 −x1 + x2 + 2x3 = 2 1 0 −1 31 1 −1 1 2 1 0 −1 −1 0 R R1 −2 −3− 0 1 1 1 − −→ 4 − R3 + R1 22 01 1 10 0 0 −1 R1 + 3 − − R→ 0 1 1 4 − 1 −− R2 −4R3 0 1 −1 00 3 1 −3 x1 −1 1 =⇒ x = x2 = 7 = 7 3 3 1 x3 −1 − 10 31 A|b = −1 1 1 R3 −R2 − − → 0 −− − 1 R3 3 0 x1 x2 x3 = 0 1 2 . 0 1 2 0 −1 3 7 0 3 1 1 −3 3 5 / 13 3 Example 2 Equivalent to: x1 + 2x2 − x3 = 0 2x1 + x2 + x3 = 0 x1 − x2 + 2x3 = 0 1 2 −1 211 1 −1 2 x1 x2 x3 = 0 0 0 . This system is homogeneous: it’s equivalent to the matrix equation Ax = 0. Since x = 0 is always a solution, the only possibilities are (1) a unique solution or (2) infinitely many solutions. 1 2 −1 0 1 2 −1 0 R2 −2R1 1 1 0 − − − 0 −3 3 0 − −→ A | 0 = 2 R3...
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