Solving System Of Linear Equations 线性方程组

1. Solving System Of Linear Equations 线性方程组 二元一次方程组

2. (2’) -(1) 2(2) Solve the following system of linear equations: 1. Elimination加减消元法Substitution代入消元法

3. 2. Gaussian Elimination 消元法 z = 1, y = 1, x = 4 Alternative

4. Alternative – Elementary row transformations Coefficient Matrix 系数矩阵 Augmented Matrix 增广矩阵 Algorithm Practice z = 1, y = 1, x = 4 • Interchanging 2 rows: RiRj • Multiply a row by a constant: RikRi • Add to a row a multiple of another row :RikRj+Ri R2  – 2R1+R2 R3  – 3R1+R2 R3  –R2 +R3 R3 0.5R3

5. ALGORITHM R1  – 3R3+R1 R2  – 2R3 +R2 R1  – 2R2 +R1 The matrix can be simplified further x = 4, y = 1, z = 1 Computer Simulation

6. PRACTICE 1. 2. In general, how many solution(s) can a system of linear equations have? 3. BACK Solution

7. 1. 3. 2. SOLUTION BACK 0z=2 no solution x = (1+)/2, y = 4x – 3 z = 2 –  where IR z=,

8. Computer program 1. Solving linear equations in 3 variables • http://www.mkaz.com/math/js_lalg3.html 2. Step by Step Illustration All steps shownhttp://www1.minn.net/~dchristo/GElim/GElim.html With final scorehttp://wims.unice.fr/wims/wims.cgi?lang=cn&cmd=new&module=U1%2Falgebra%2Fvisgauss.cn&type=system&size=3&field=Q Further points of discussion • How can the ALGORITHM described above be applied to • 4 linear equations in 4 variables? • n linear equations in n variables? 2. What are the limitations of the above computer programs? Websiteshttp://cos.cumt.edu.cn/math/shuxuekejian/xianxingdaishu/charp1/124.htm http://ws1.hkcampus.net/~ws1-kcy/al_pmath.html

9. A Mathematical Problem Amy, Ben and Calvin play a game as follows. The player who loses each round must give each of the other players as much money as the player has at that time. In round 1, Amy loses and gives Ben and Calvin as much money as they each have. In round 2, Ben loses, and gives Amy and Calvin as much money as they each then have. Calvin loses in round 3 and gives Amy and Ben as much money as they each have. They decide to quit at this point and discover that they each have $24. How much money did they each start with? Approach: (1) top down strategy (2) bottom up strategy

10. Method 1 – Top down strategy x – y – z 2y 2z 4z 2(x – y – z) 2(3y – x – z) 4(x – y – z) Let Amy, Ben and Calvin each had $x, $y and $z initially. 3y – x – z 7z – x – y x – y – z = 6, 3y – x – z = 12 and 7z – x – y = 24; from which x, y and z can be solved. x=39, y=21, z=12 NEXT

11. Method 2 – Bottom up strategy Instead of considering how much money Amy, Ben and Calvin had originally, work backwards from the moment when each of them has $24 each. Complete the following table and see how easily you can reach exactly the same conclusion as in method 1. BACK