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ch1 - McGill University Math 270 Applied Linear Algebra...

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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 0 , otherwise, called non-linear 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) 0-0
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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 3D-space: ( 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; 0-1
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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 fl fl fl fl fl fl fl fl fl b 1 b 2 . . . b n 0-2
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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. When the above operations applied on the system, the sys- tem coefficients and constants are changed. Accordingly, the following operations are performed on the augmented matrix A # : Interchange rows: R i ⇐⇒ R j ; Multiply a row by a nonzero constant: R i ⇐⇒ kR i ; 0-3
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Add a multiple of one row to another row: R i ⇐⇒ R i + kR j Example 1 : 1 2 4 2 - 5 3 4 6 - 7 fl fl fl fl fl fl 2 6 8 , x
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