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# Row Reducing

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• 1.2C
• ### Row Reducing

1. 1. Row Reducing<br />How to get to Reduced Row Echelon Form<br />
2. 2. Strategies<br /><ul><li> Use rows with leading 1’s
3. 3. Get the leading 1’s starting at the pivot
4. 4. Use the pivot to get zeros below</li></ul> and/or above<br /><ul><li> Use rows with zeros on top of each</li></ul> other as you move right<br />
5. 5. Elementary Row Operations<br />Allowable Row Operations<br />Notation<br />Ri↔Rj<br /><ul><li> Interchange two rows
6. 6. Multiply a row through</li></ul> by a non-zero constant<br /><ul><li> Add a multiple of one</li></ul> row to another. <br />kRi->Ri<br />kRi + Rj-> Rj<br />
7. 7. EXAMPLE I<br />PIVOT<br />Use this one, called the pivot, to get zeros below.<br />-3R1 + R2-> R2<br />R1 + R3-> R3<br />
8. 8. EXAMPLE I<br />-3R1 + R2-> R2<br />
9. 9. EXAMPLE I<br />R1 + R3-> R3<br />
10. 10. EXAMPLE I<br />New pivot location<br />We want a 1 here but we also want to avoid fractions<br />-2R3 + R2-> R2<br />
11. 11. EXAMPLE I<br />New pivot <br />Use this new pivot to get zeros above and below<br /> R2 + R1-> R1<br />-2R2 + R3-> R3<br />
12. 12. EXAMPLE I<br />New pivot location <br />Now get a 1 here.<br /> R3-> R3<br />
13. 13. EXAMPLE I<br />New pivot<br />Use this new pivot to make zeros above.<br />11R3 + R1-> R1<br />13R3 + R2-> R2<br />
14. 14. EXAMPLE I<br />
15. 15. EXAMPLE II<br />Solve the system: <br />There are two equations and four unknowns so we need a 2 x 5 matrix.<br />Augmented matrix:<br />
16. 16. EXAMPLE II<br />-2R1 + R2-> R2<br /> R2-> R2<br />←RREF<br />
17. 17. EXAMPLE II<br />Expressing the solution: <br />If a system is consistent, any column that does not contain a leading one will have a parameter.<br />Parameters required <br />for x3 and x4<br />
18. 18. EXAMPLE II<br />Expressing the solution: <br />LET<br /> s t<br />
19. 19. EXAMPLE II<br />Converting back to equations in terms of s and t: <br />Solving for x1 and x2 leads to the final solution: <br />
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