This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: McGill University Math 270: Applied Linear Algebra CHAPTER 1: SOLUTIONS OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 1 Introduction The following chapters will devote the study of linear algebraic equations . A equation is generally written in the form, like f ( x ) = 0 . • It is called algebraic equation , if the function f ( x ) is polynomial, otherwise, called transcendental equation , • It is called Linear equation , if f ( x ) = a 1 x 1 + a 2 x 2 + ··· + a , otherwise, called nonlinear equation . In this chapter, we are going to find the solution properties of a system of ( m × n ) linear algebraic equation, such as ( m = 2 ,n = 3) : x 1 + x 2 − x 3 = 1 , 3 x 1 + 2 x 2 − 4 x 3 = 3 . (1) 00 2 Solutions for general ( m × n ) System of Linear Equations In general, a system of ( m × n ) equation is a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = b 1 , a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = b 2 , . . . a m 1 x 1 + a m 2 x 2 + ··· + a mn x n = b m . As m = n = 3 , we have the system of equations: a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 , a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 , a 31 x 1 + a 32 x 2 + a 3 n x 3 = b 3 . (2) Here, each of the equations may be described by a plane in the 3Dspace: ( x 1 ,x 2 ,x 3 ) . Hence, a solution of system (2) corresponds an intersection point of these three planes. There are four possibilities regarding to the intersection points of the planes: • These planes have no intersection point, so the system has no solution; 01 • These planes have just one intersection point, so the system has unique solution; • These planes have a intersection line, so the system has infinitely many solutions with an arbitrary constant; • These planes are coincident, so the system has infinitely many solutions with two arbitrary constant. For a general n dimensional case, we have similar situation. In the following subsections, we are going to develop a system atic procedure for solving a system of equations. In doing so, one often starts with the following forms of matrices: 1. The matrix of coefficients: A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . a m 1 a m 2 ··· a mn 2. The augmented matrix of coefficients: A # = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . a m 1 a m 2 ··· a mn b 1 b 2 . . . b n 02 2.1 (*)Elementary Row Operations Before we solve a given ( m × n ) system, let us consider the following question: what types of operations can be applied on such a system without altering its solution set. The following three elementary operations are, obviously, of these types: • Interchange equations; • Multiply an equation by a nonzero constant; • Add a multiple of one equation to another equation....
View
Full
Document
This note was uploaded on 11/22/2010 for the course MIME MIME 310 taught by Professor Jassim during the Winter '09 term at McGill.
 Winter '09
 Jassim

Click to edit the document details