2130_Chapter7_Notes

Since x 0 is always a solution the only possibilities

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online