examples152-ch6

examples152-ch6 - MATH 152 Fall 2006-07 Applied Linear...

Info iconThis preview shows pages 1–7. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 152 Fall 2006-07 Applied Linear Algebra & Differential Equations Worked Examples Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology September 1, 2006 ii Contents Table of Contents iii 1 Introduction 1 2 First-Order Differential Equations 5 3 Second-Order Linear Equations 39 4 Laplace Transform 83 5 Matrix 127 6 Systems of Linear Equations 135 7 Euclidean Vector 155 8 Eigenvalue and Eigenvector 177 9 Systems of Differential Equations 209 10 Orthogonality 231 iii Chapter 6 Systems of Linear Equations ¥ Example 6.1 (Augmented matrix) Write down the augmented matrix for the system of linear equations. 8 < : x 1- x 3 = 5 , x 1 + x 2 + x 4 = 1 , x 1- 3 x 4 = 2 . Solution The augmented matrix is 2 4 1- 1 5 1 1 1 1 1- 3 2 3 5 . 2 ¥ Example 6.2 (Augmented matrix) Suppose ax + by = 1 represents a straight line which passes through the points (1 , 2) and (3 , 4) . Write down the augmented matrix for the corresponding system of linear equations for a , b . Solution For ax + by = 1 to pass through (1 , 2) and (3 , 4), we have a · 1 + b · 2 = 1 and a · 3 + b · 4 = 1. Then we have two linear equations with two variables a , b . The augmented matrix is » 1 2 1 3 4 1 – . 2 ¥ Example 6.3 (Row echelon form) Which of the following matrices are in row echelon form? A = » 3 4 – , B = 2 4 1 1 1 1 3 5 , C = 2 4 3 1 3 3 5 , P = 2 4 1 1 1 3 5 , Q = 2 4 3 5 , R = 2 4 1 1 1 1 3 5 , X = »- 1 1 1 – , Y = » 1- 1 1- 1 – , Z = 2 4 1- 1- 1 1 3 5 . Solution Matrices A , C , P , Q , X , Y , Z are in row echelon form. Matrix B is not in row echelon form because the second row which consists of all zero entries is not at the bottom of the matrix. Matrix R is not in row echelon form because the (2 , 1)-entry is not zero. 2 135 6. Systems of Linear Equations ¥ Example 6.4 (Gaussian elimination – unique solution) Solve the system of linear equations 8 > < > : x 1 + x 2- 2 x 3 = 1 , 2 x 1 +3 x 2- 2 x 3 =- 2 , 3 x 1- 11 x 3 = 4 . Solution We simplify the augmented matrix as follows 2 4 1 1- 2 1 2 3- 2- 2 3- 11 4 3 5- 2 R 1 + R 2------→- 3 R 1 + R 3 2 4 1 1- 2 1 1 2- 4- 3- 5 1 3 5 3 R 2 + R 3-----→ 2 4 1 1- 2 1 1 2- 4 1- 11 3 5 . Interpreted the last matrix as a system of linear equations, we have 8 < : x 1 + x 2- 2 x 3 = 1 , x 2 +2 x 3 =- 4 , x 3 =- 11 . By backward substitution , starting from the third equation, we have x 3 =- 11. Substituting this value into the second equation, we have x 2 = 18. Substituting the values of x 2 and x 3 into the first equation, we have x 1 =- 39. We conclude that the system has a unique solution ( x 1 ,x 2 ,x 3 ) = (- 39 , 18 ,- 11). 2 ¥ Example 6.5 (Gaussian elimination – unique solution) Solve the system of linear equations 8 > < > : x 1 +2 x 2 = 7 , 2 x 1- 3 x 2 =- 7 , 3 x 1- 5 x 2 =- 12 ....
View Full Document

Page1 / 24

examples152-ch6 - MATH 152 Fall 2006-07 Applied Linear...

This preview shows document pages 1 - 7. Sign up to view the full document.

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