lecture14 - Differential Equations & Linear...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Differential Equations & Linear Algebra Lecture 14 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Homogeneous systems Contents Homogeneous systems Non-singular matrices Example Example Solve the system 6 x 3 + 2 x 4- 4 x 5- 8 x 6 = 8 3 x 3 + x 4- 2 x 5- 4 x 6 = 4 2 x 1- 3 x 2 + x 3 + 4 x 4- 7 x 5 + x 6 = 2 6 x 1- 9 x 2 + 11 x 4- 19 x 5 + 3 x 6 = 1 . Solution The augmented matrix is A = 0 6 2- 4- 8 8 0 3 1- 2- 4 4 2- 3 1 4- 7 1 2 6- 9 0 11- 19 3 1 which is row equivalent to Example(cont.) B = 1- 3 2 11 6- 19 6 1 24 0 1 1 3- 2 3 5 3 0 0 0 1 1 4 0 0 0 0 . The complete solution is x 1 = 1 24 + 3 2 x 2- 11 6 x 4 + 19 6 x 5 , x 3 = 5 3- 1 3 x 4 + 2 3 x 5 , x 6 = 1 4 , with x 2 , x 4 , x 5 arbitrary. (Here n = 6 , r = 3 , c 1 = 1 , c 2 = 3 , c 3 = 6; c r < n + 1; r < n. ) Exercise Exercise For which numbers a and b does the following system have (i) no solution, (ii) a unique solution, (iii) infinitely many solutions. x- 2 y + 3 z = 4 2 x- 3 y + az = 5 3 x- 4 y + 5 z = b. Solution The augmented matrix A is row equivalent to B = 1- 2 3 4 1 a- 6- 3- 2 a + 8 b- 6 . Exercise(cont.) Case 1. a 6 = 4 , then B can be reduced to the simplest form 1 0 0 u 0 1 0 v 0 0 1 w , and we have the unique solution. Case 2. a = 4 , then B = 1- 2 3 4 1- 2- 3 b- 6 . When b 6 = 6 , we get no solution, whereas if b = 6 , then B is row equivalent to 1 0- 1- 2 0 1- 2- 3 0 0 . We obtain the complete solution x =- 2 + z, y =- 3 + 2 z . Homogeneous systems A system of homogeneous linear equations is a system of the form a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = 0 ....
View Full Document

This note was uploaded on 05/17/2011 for the course ME 255 taught by Professor Jianlixie during the Fall '10 term at Shanghai Jiao Tong University.

Page1 / 23

lecture14 - Differential Equations & Linear...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online