 # Chap. 1 Systems of Linear Equations

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Chap. 1 Systems of Linear Equations
1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination and Gauss-Jordan Elimination 1.3 Applications of Systems of Linear Equations

Chapter Objectives Recognize, graph, and solve a system of linear equations in n variables. Use back substitution to solve a system of linear equations. Determine whether a system of linear equations is consistent or inconsistent. Determine if a matrix is in row-echelon form or reduced row-echelon form. Use element row operations with back substitution to solve a system in row-echelon form. Use elimination to rewrite a system in row echelon form. Ming-Feng Yeh Chapter 1

Chapter Objective (cont.)
Write an augmented or coefficient matrix from a system of linear equation, or translate a matrix into a system of linear equations. Solve a system of linear equations using Gaussian elimination with back-substitution. Solve a homogeneous system of linear equations. Set up and solve a system of linear equations to fit a polynomial function to a set of data points, as well as to represent a network. Ming-Feng Yeh Chapter 1

1.1 Introduction Linear Equations in n Variables
A linear equations in n variables x1, x2, …, xn has the form: a1x1 + a2x2 + … + anxn = b Coefficients: a1, a2, …, an  real number Constant term: b  real number Leading Coefficient: a1 Leading Variable: x1 Linear equations have no products or roots of variables and no variables involved in trigonometric, exponential or logarithmic functions. Variables appear only to the first power. Ming-Feng Yeh Chapter 1

Example 1 Linear Equations: Nonlinear Equations: Product of variables
Section 1-1 Example 1 Linear Equations: Nonlinear Equations: True? Product of variables involved in exponential involved in trigonometric Not the first power Ming-Feng Yeh Chapter 1

Example 2 Parametric Representation of a Solution Set
Section 1-1 Example 2 Parametric Representation of a Solution Set Solve the linear equation x1 + 2x2 = 4 Sol: x1 = 4  2x2 Variable x2 is free (it can take on any real value). Variable x1 is not free (its value depends on the value of x2). By letting x2 = t (t: the third variable, parameter), you can represent the solution set as 參數個數 = 變數個數  方程式列數 Ming-Feng Yeh Chapter 1

Infinite number of solutions
Section 1-1 Example 3 Parametric Representation of a Solution Set Solve the linear equation 3x + 2y  z = 3 Sol: Choosing y and z to be the free variables Letting y = s and z = t, you obtain the parametric representation Infinite number of solutions Ming-Feng Yeh Chapter 1

Systems of Linear Equations
Section 1-1 Systems of Linear Equations A system of m linear equations in n variables is a set of m equations, each of which is linear in the same n variables: The double-subscript notation indicates that aij is the coefficient of xj in the ith equation. A system of linear equations has exactly one solution, an infinite number of solutions, or no solution. A system of linear equations is called consistent if it has at least one solution and inconsistent if it has no solution. Ming-Feng Yeh Chapter 1

Example 4 Systems of two equations in two variables
Section 1-1 Example 4 Systems of two equations in two variables Solve the following systems of linear equations, and graph each system as a pair of straight lines. y x Two intersecting lines x y x y Two coincident lines Two parallel lines Ming-Feng Yeh Chapter 1

Solving a system of linear equations
Section 1-1 Solving a system of linear equations Row-echelon form: it follows a stair-step pattern and has leading coefficient of 1. Using back-substitution to solve a system in row-echelon form. Example 6. 1. From Eq. 3 you already know the value of z. 2. To solve for y, substitute z = 2 into Eq. 2 to obtain y = 1. 3. Substitute z = 2 and y = 1 into Eq. 1 to obtain x = 1. Ming-Feng Yeh Chapter 1

Section 1-1 Equivalent Systems Two systems of linear equations are called equivalent if they have precisely the same solution set. Each of the following operations on a system of linear equations produces an equivalent system. 1. Interchange two equations. 2. Multiply an equation by a nonzero constant. 3. Add a multiple of an equation to another equation. Gaussian elimination: Rewriting a system of linear equations in row-echelon form usually involves a chain of equivalent systems, each of each is obtained by using one of the three basic operations. Ming-Feng Yeh Chapter 1

Example 7 A system with exactly one solution Solve the system
Section 1-1 Example 7 A system with exactly one solution Solve the system Adding the first equation to the second produces a new second equation. Adding –2 times the first equation to the third equation produces a new third equation. Adding the second equation to the third equation produces a new third equation. (2)  (1/2) The solution is x = 1, y = 1, and z = 2. Ming-Feng Yeh Chapter 1

Example 8 An Inconsistent System Solve the system (2)
Section 1-1 Example 8 (2) An Inconsistent System Solve the system Adding –2 times the first equation to the second produces a new second equation. Adding –1 times the first equation to the third produces a new third equation. Adding –1 times the 2nd equation to the 3rd produces a new 3rd equation. (1) (1) Because the third “equation” is a false statement, this system has no solution. Moreover, because this system is equivalent to the original system, you can conclude that the original system also has no system. Ming-Feng Yeh Chapter 1

Example 9 A system with an infinite number of solutions
Section 1-1 Example 9 A system with an infinite number of solutions Solve the system The first two equations are interchanged. Adding the first equation to the third produces a new third equation. Adding –3 times the 2nd equation to the 3rd produces a new 3rd equation. (3) unnecessary Let x3 = t, t  R Ming-Feng Yeh Chapter 1

1.2 Gaussain Elimination and Gauss-Jordan Elimination
Definition: Matrix If m and n are positive integers, then an mn matrix is a rectangular array in which each entry, aij, of the matrix is a number. An mn matrix has m rows (horizontal lines) and n columns (vertical lines). The entry aij is located in the ith row and the jth column. A matrix with m rows and n columns (an mn matrix) is said to be of size mn. If m = n, the matrix is called square of order n. For a square matrix, the entries a11, a22, a33, … are called the main diagonal entries. Ming-Feng Yeh Chapter 1

Augmented/Coefficient Matrix
Section 1-2 Augmented/Coefficient Matrix The matrix derived from the coefficients and constant terms of a system of linear equations is called the augmented matrix of the system. The matrix containing only the coefficients of the system is called the coefficient matrix of the system. System Augmented Matrix Coefficient Matrix x y z const. Ming-Feng Yeh Chapter 1

Elementary Row Operations
Section 1-2 Elementary Row Operations Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row. Two matrices are said to be row-equivalent if one can be obtained from the other by a finite sequence of elementary row operations. Ming-Feng Yeh Chapter 1

Example 3 Using Elementary Row Operation to Solve a System
Section 1-2 Example 3 Using Elementary Row Operation to Solve a System Linear System Associated Augmented matrix R2+R1R2 R3+(2)R1R3 (2) R3+R2R3 0.5R3R3 0.5 Ming-Feng Yeh Chapter 1

Row-Echelon Form of a Matrix
Section 1-2 Row-Echelon Form of a Matrix A matrix in row-echelon form has the following properties. All rows consisting entirely of zeros occur at the bottom of the matrix. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). For two successive (nonzero) rows, the leading 1 in the higher row is father to the left than the leading 1 in the lower row. Remark: A matrix in row-echelon form is in reduced row-echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1. Ming-Feng Yeh Chapter 1

Example 4 In row-echelon form Not in row-echelon form Section 1-2
Ming-Feng Yeh Chapter 1

Gaussian Elimination with Back-Substitution
Section 1-2 Gaussian Elimination with Back-Substitution Write the augmented matrix of the system of linear equations. Use elementary row operations to rewrite the augmented matrix in row-echelon form. Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution. Ming-Feng Yeh Chapter 1

Example 5 A system with exactly one solution Solve the system (2)
Section 1-2 Example 5 A system with exactly one solution Solve the system (2)  6 Ming-Feng Yeh Chapter 1

Example 6 A system with no solution Solve the system (1) (3) (2)
Section 1-2 Example 6 A system with no solution Solve the system (3) (1) (2) 0 = 2 … ??? The original system of linear equations is inconsistent. Ming-Feng Yeh Chapter 1

Gauss-Jordan Elimination
Section 1-2 Gauss-Jordan Elimination Continues the reduction process until a reduced row-echelon form is obtained. Example 7: Use Gauss-Jordan elimination to solve the system In Ex. 3  2 (3) (9) Ming-Feng Yeh Chapter 1

Example 8 A System with an Infinite Number of Solutions
Section 1-2 Example 8 A System with an Infinite Number of Solutions Solve the system of linear equations Let x3 = t, t  R Ming-Feng Yeh Chapter 1

Homogeneous Systems of Linear Equations
Section 1-2 Homogeneous Systems of Linear Equations Each of the constant terms is zero. A homogeneous system must have at least one solution. Trivial (obvious) solution: all variables in a homogeneous system have the value zero, then each of the equation must be satisfied. Ming-Feng Yeh Chapter 1

Example 9 Solve the system of linear equations Let x3 = t, t  R
Section 1-2 Example 9 Solve the system of linear equations Let x3 = t, t  R Ming-Feng Yeh Chapter 1

Section 1-2 Theorem 1.1 The Number of Solutions of a Homogeneous System Every homogeneous system of linear equations is consistent. Moreover, if the system has fewer equations than variables, then it must have an infinite number of solutions. Ming-Feng Yeh Chapter 1

1.3 Applications of Systems of Linear Equations
Polynomial Curve Fitting fit a polynomial function to a set of data points in the plane. n points: Polynomial function: Network Analysis focus on networks and Kirchhof’s Laws for electricity. Ming-Feng Yeh Chapter 1

Section 1-3 Network Analysis Networks composed of branches and junctions are used as models in the fields as diverse as economics, traffic analysis, and electrical engineering. The total flow into a junction is equal to the total flow out of the junction. Example. Ming-Feng Yeh Chapter 1

Section 1-3 Example 5 Ming-Feng Yeh Chapter 1

Section 1-3 Kirchhoff’s Laws All the current flowing into a junction must flow out of it. (KCL) The sum of the products IR (I is the current and R is the resistance) around a closed path is equal to the total voltage in the path. (KVL) A closed path is a sequence of branches such that the beginning point of the first branch coincides with the end point of the last branch. Ming-Feng Yeh Chapter 1

Section 1-3 Example 6  or : Path 1: Path 2: Ming-Feng Yeh Chapter 1

Example 7 See pp. 37-38 in the textbook Section 1-3 Ming-Feng Yeh
Chapter 1