# 1 Systems Of Linear Equations and Matrices

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 1 Systems Of Linear Equations and Matrices 1.1 Systems Of Linear Equations In this section you ll learn what Systems Of Linear Equations are and how to solve them. Remember that equations of the form a 1 x + a 2 y = b, for a 1, a 2 R\{0}, b R describe lines in a 2-dimensional (x y) coordinate system. This is called a linear equation in x and y. More generally Definition 1 A linear equation in n variables, x 1, x 2,..., x n, is given by where a 1, a 2,..., a n, b R. a 1 x 1 + a 2 x a n x n = b Example 1 3x 1 + 2x 2 = 9, 3x 1 2x 2 + x 3 = 25 are linear. But 3x x 2 = 9 and 3x 1 2x 2 + x 3 = 25 are not linear. They have x 2 1, and x 3 included which are non linear terms in the variables. Definition 2 1. A set of linear equations in the variables x 1, x 2,..., x n is called a system of linear equations or a linear system. 2. A sequence of numbers s 1, s 2,... s n is called a solution of the system if x 1 = s1, x 2 = s 2,..., x n = s n is a solution of every equation in the system. Example 2 1. Consider the following linear system 3x 1 + 2x 2 3x 3 = 10 Then x 1 = 3, x 2 = 2, x 3 = 1 is a solution of the linear system, because 3(3) + 2(2) 3(1) = = 2 However x 1 = 1, x 2 = 1, x 3 = 1 is not a solution, because 3(1) + 2(1) 3(1) =

2 2. Not all linear systems have solutions For example 3x 1 + 3x 2 = 9 3x 1 + 3x 2 = 3 does not have a solution, the 2 equations contradict each other. 3. Other linear systems have infinite many solutions 3x 1 + 3x 2 = 9 x 1 + x 2 = 3 For all t R x 1 = t and x 2 = 3 t are solutions of the linear system. Check: 3t + 3(3 t) = 3t + 9 3t = 9 t + (3 t) = t + 3 t = 3 To find just one solution, e.g. choose x 1 = t = 3, x 2 = 3 t = 3 3 = 0, then this is one solution of the linear system. Geometrically in 2 dimensions: We will later prove that for every linear system exactly one of three possibilities hold 1. has no solution 2. has exactly one solution, or 3. has infinitely many solutions Definition 3 A linear system that has no solution is called inconsistent A linear system with at least one solution is called consistent. 2

3 In order to abbreviate the writing (mathematicians are lazy, when it comes to writing) matrices are used to represent linear systems. Example 3 The following linear system 3x 1 + 2x 2 3x 3 = 10 4x 1 + 2x 2 = 16 can be represented, by just listing the constants in the system, but the location has to be kept in mind. The augmented matrix representing this linear system is In general: An arbitrary system of m linear equations in n unknowns can be written as a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b a m1 x 1 + a m2 x a mn x n = b m where x 1, x 2,..., x n are the unknowns, and the a s and b s denote the constants. The indexes on the coefficients are specifying the location in the system. The first index indicates the equation, and the second index gives the unknown the coefficient multiplies. a 53 is the multiplier for variable x 3 in equation 5. Definition 4 (1) 1. A matrix is a rectangular array of numbers. 2. The augmented matrix for the linear system (6) is a 11 a a 1n b 1 a 21 a a 2n b a m1 a m2... a mn b m (2) The first task is to learn, how to solve linear equations. You will learn an algorithm to do so. This is very essential to be mastered to succeed in this course. We will use the algorithm over and over for solving different types of problems. Systems of linear equations are solved by manipulating the system through operations, that do not change the solution, but make finding the solution easier. Elementary row operations: 3

4 1. Multiply an equation by a constant (not zero). 2. Interchange two equations. 3. Add a multiple of one equation to another. Use the operations to eliminate unknowns from equations to make solving the system easier. To save some writing we could do the same operations with the augmented matrix Example 4 The linear system and its augmented matrix 3x 1 + 2x 2 3x 3 = 10 4x 1 + 2x 2 = Add -3 times the first equation to the second (1st row) + (2nd row) 5x 2 6x 3 = x 1 + 2x 2 = Add -4 times the first equation to the third (1st row) + (3rd row) 5x 2 6x 3 = x 2 4x 3 = Multiply the second equation by 1/5 3. 1/5 (2nd row) x 2 6/5x 3 = 4/ /5 4/5 6x 2 4x 3 = Add -6 times the second equation to the third (2nd row) +(3rd row) x 2 6/5x 3 = 4/ /5 4/5 16/5x 3 = 16/ /5 16/5 5. Multiply the third equation by 5/ /16 (3rd row) x 2 6/5x 3 = 4/ /5 4/5 x 3 = Add -1 times the third equation to the first 6. -(3rd row)+ (1st row) x 1 x 2 = x 2 6/5x 3 = 4/ /5 4/5 x 3 =

5 7. Add -6/5 times the third equation to the second 7. -6/5(3rd row)+ (2nd row) x 1 x 2 = x 2 = 10/ x 3 = Add 1 times the second equation to the first 8. (2nd row)+ (1st row) x 1 = x 2 = x 3 = Now it is obvious that the solution is x 1 = 3, x 2 = 2, x 3 = 1. This system has exactly one solution. Example 5 1. Assume you have one equation in 3 variables 3x + 3y 3z = 9 What are the solutions of this equation? Since we have only one equation, but three variables, we can choose freely (3-1) variables. Let s, t R, then y = s, z = t, 3x + 3s 3t = 9 y = s, z = t, x = (9 3s + 3t)/3 = 3 s + t is a solution for the equation. E.g. choose y = 1, z = 1, then with x = = is one possible solution. 2. x 1 2x 2 = 5 2x 1 + 4x 2 = 10 Use the augmented matrix to solve this linear system [ ] [ (1st row)+(2nd row) ] The last row translates to 0 = 0, which is always true, and gives no restriction on the unknowns solving the system. This line can be ignored when determining the solution. (2 variables - 1 equation) For s R, x 2 = s, x 1 = 5 + 2s is a solution of the linear system above. The system has infinite many solutions. 3. x 1 2x 2 = 5 2x 1 + 4x 2 = 5 Use the augmented matrix to solve this linear system [ ] [ (1st row)+(2nd row) ] The last row translates to 0 = 15, which is not true. No choice of the variables will lead to a solution of the linear system. This system has no solution. 5

6 1.2 Gaussian Elimination Definition 5 1. A matrix is in row-echelon form, if and only if (a) The leading coefficient of each nonzero row is one. (leading 1) (b) All rows consisting entirely of zeros are grouped at the bottom of the matrix. (c) The leading nonzero coefficient of a row is always strictly to the right of the leading coefficient of the row above it. 2. A matrix is in reduced row-echelon form, if and only if the matrix is in row-echelon form, and (d) each column that contains a leading 1 has zero everywhere else in that column. Theorem 1 Every matrix in reduced row-echelon form is also in row-echelon form. Proof: Assume A is any matrix in reduced row-echelon form. Therefore properties (a)-(d) from Definition 5 hold for A. Therefore particularly properties (a)-(c) hold for A. Therefore according to the definition A is in row-echelon form. End of proof. (Use therefore = ) Gaussian Elimination is an algorithm to transform a matrix into a equivalent matrix in row-echelon form. Gauss-Jordan elimination is an algorithm to transform a matrix into a equivalent matrix in reduced row-echelon form. Example 6 1. Gives solution x 1 = 5, x 2 = 2, x 3 = [ x 1 and x 2 correspond to leading 1s in the matrix, they are called pivots or leading variables. The nonleading variables are called free, here x 3. Free variables can be assigned arbitrary value from R, say s. Then the leading variables can be solved in s. This gives solutions x 1 = 5, x 2 = 2 2s, x 3 = s, for any s R. 6 ]

7 The last equation translates to 0x 3 = 10, so 0 = 10. This can not be satisfied, the linear system has no solution. Gaussian Elimination step 1 Locate the leftmost column that is not entirely 0. step 2 Exchange the top row with another row, if necessary, to bring a non zero entry to the top of the column from step 1. step 3 If the entry is not one, say it is a, then multiply the row by 1/a. step 4 Add suitable multiples of the top row to the rows bellow so that all entries below the leading one become zeros. step 5 Now start over with step 1 for the matrix excluding row 1. The matrix is now in row echelon form Gauss-Jordan Elimination Step 1 to Step 5, and step 6 Beginning with the last nonzero row and working upward, add suitable multiples of each row to the rows above to introduce zeros above the leading 1s. After step 6 the matrix is in reduced row-echelon form. Example 7 Consider a linear system with the following augmented matrix step 1 First column step step 3 the entry is 1 7

8 step 4-2(row 1)+(row 3) step 5 start over with step 1. step 1 second column step 2 it is not zero step 3-1(row 2) step 4 2(row 2)+ (row 3) step 5 start over with step 1 step 1 third column step 2 1/2(row 3) This ends the Gaussian elimination, the matrix is in row-echelon form. Using back substitution we can use the linear system from here. The third row translates to x 3 = 9. The second row translates to x 2 + 2(x 3 ) = 16, using x 3 = 9, we get x 2 = 16 2(9) = 2 The first row translates to x 1 + 2x 2 + x 3 = 8, with x 2 = 2, x 3 = 9, we get x 1 = 8 2( 2) 9 = 3. The only solution is x 1 = 3, x 2 = 2, x 3 = 9. Instead of using back substitution we could have found the reduced row-echelon form by using step 6. step 6-2(row 3) + row 2, -1(row3)+(row1)

9 -2(row 2)+ (row 1) This is the reduced row-echelon form, which translate immediately into the solution x 1 = 3, x 2 = 2, x 3 = 9. Definition 6 A linear system is said to be homogeneous if in b 1 = b 2 =... = b m = 0 a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b a m1 x 1 + a m2 x a mn x n = b m Theorem 2 All homogeneous linear systems have at least the trivial solution x 1 = x 2 =... = x n = 0. Proof: Substitute x 1 = x 2 =... = x n = 0 into the first equation, and find a a a 1n 0 = 0 Equation 1 is satisfied. In the same way we see that all the other equations will be satisfied, and x 1 = x 2 =... = x n = 0 is a solution of the homogeneous system. Definition 7 All solutions different from x 1 = x 2 =... = x n = 0 are called nontrivial. Theorem 3 Every homogeneous linear system with more variables than equations has infinite many solutions. Proof:(not presented in class) m < n Then the reduced row echelon form of the augmented matrix for the linear system looks like 1 S S S S m S m 0 Where S j are the coefficients for the n m > 0 free variables in row j, 1 j m. 9

10 Most important to observe, there have to be free variables, which can be set to arbitrary numbers in R, and each choice results in a different solution, so that there are must be infinite many different solutions. Solutions of homogeneous linear systems, can also be found using Gaussian or Gauss-Jordan Elimination Example 8 The augmented matrix of a homogeneous linear system is (r1)+r2, 3(r1)+r /6(r2) (r2)+r (backward substitution) The third row only yields 0 = 0, from the second row we get x 2 = x 3, from the first row, we then get x 1 + 2x 3 3x 3 = 0 x 1 = x 3. For any s R let x 3 = s (free variable), then x 1 = x 2 = x 3 = s is a solution of the linear system. With s = 0, we get the trivial solution. 10

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Linear Equations in Linear Algebra

1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero

### Row Echelon Form and Reduced Row Echelon Form

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Reduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:

Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in

### Systems of Linear Equations

A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Section 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)

Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix

### 4 Solving Systems of Equations by Reducing Matrices

Math 15 Sec S0601/S060 4 Solving Systems of Equations by Reducing Matrices 4.1 Introduction One of the main applications of matrix methods is the solution of systems of linear equations. Consider for example

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Solving Systems of Linear Equations; Row Reduction

Harvey Mudd College Math Tutorial: Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study The method reviewed

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### 2.5 Gaussian Elimination

page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3

### Linear Systems and Gaussian Elimination

Eivind Eriksen Linear Systems and Gaussian Elimination September 2, 2011 BI Norwegian Business School Contents 1 Linear Systems................................................ 1 1.1 Linear Equations...........................................

### Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### MAT Solving Linear Systems Using Matrices and Row Operations

MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented

### APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A.

APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the co-factor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj

### 1.5 SOLUTION SETS OF LINEAR SYSTEMS

1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as

### Lecture Notes 2: Matrices as Systems of Linear Equations

2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Math 240: Linear Systems and Rank of a Matrix

Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011

### Systems of Linear Equations

Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11-]. Main points in this section: 1. Definition of Linear

### Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### Linear Equations ! 25 30 35\$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development

MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!

### Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

### Section 6.2 Larger Systems of Linear Equations

Section 6.2 Larger Systems of Linear Equations EXAMPLE: Solve the system of linear equations. Solution: From Equation 3, you know the value z. To solve for y, substitute z = 2 into Equation 2 to obtain

### 5 Homogeneous systems

5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m

### Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION

8498_CH08_WilliamsA.qxd 11/13/09 10:35 AM Page 347 Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION C H A P T E R Numerical Methods 8 I n this chapter we look at numerical techniques for

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### ( % . This matrix consists of \$ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### Chapter 2. Linear Equations

Chapter 2. Linear Equations We ll start our study of linear algebra with linear equations. Lots of parts of mathematics arose first out of trying to understand the solutions of different types of equations.

### 2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

### Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns

Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in

### 5.5. Solving linear systems by the elimination method

55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve

### Matrices, Determinants and Linear Systems

September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we

### SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

### Solving Linear Systems, Continued and The Inverse of a Matrix

, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Solving a System of Equations

11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### 2. Perform elementary row operations to get zeros below the diagonal.

Gaussian Elimination We list the basic steps of Gaussian Elimination, a method to solve a system of linear equations. Except for certain special cases, Gaussian Elimination is still state of the art. After

### Lecture 1: Systems of Linear Equations

MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

### Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

### Linearly Independent Sets and Linearly Dependent Sets

These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

### We seek a factorization of a square matrix A into the product of two matrices which yields an

LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable

### Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

### SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

### Chapter 4 - Systems of Equations and Inequalities

Math 233 - Spring 2009 Chapter 4 - Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011

Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,

### Direct Methods for Solving Linear Systems. Linear Systems of Equations

Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### 10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method

578 CHAPTER 1 NUMERICAL METHODS 1. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after

### Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

### Linear Equations in Linear Algebra

1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A 0, where A

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 4. Gaussian Elimination In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor

### 8.3. Gauss elimination. Introduction. Prerequisites. Learning Outcomes

Gauss elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated electrical

### These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

### Practical Numerical Training UKNum

Practical Numerical Training UKNum 7: Systems of linear equations C. Mordasini Max Planck Institute for Astronomy, Heidelberg Program: 1) Introduction 2) Gauss Elimination 3) Gauss with Pivoting 4) Determinants

### Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).

MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix

### SECTION 8-1 Systems of Linear Equations and Augmented Matrices

86 8 Systems of Equations and Inequalities In this chapter we move from the standard methods of solving two linear equations with two variables to a method that can be used to solve linear systems with

### Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

### e-issn: p-issn: X Page 67

International Journal of Computer & Communication Engineering Research (IJCCER) Volume 2 - Issue 2 March 2014 Performance Comparison of Gauss and Gauss-Jordan Yadanar Mon 1, Lai Lai Win Kyi 2 1,2 Information

### Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### 6 Systems of Linear Equations

6 Systems of Linear Equations A system of equations of the form or is called a linear system of equations. x + y = 7 x y = 5p 6q + r = p + q 5r = 7 6p q + r = Definition. An equation involving certain

### 5.1. Systems of Linear Equations. Linear Systems Substitution Method Elimination Method Special Systems

5.1 Systems of Linear Equations Linear Systems Substitution Method Elimination Method Special Systems 5.1-1 Linear Systems The possible graphs of a linear system in two unknowns are as follows. 1. The

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### Linear Equations in Linear Algebra

May, 25 :46 l57-ch Sheet number Page number cyan magenta yellow black Linear Equations in Linear Algebra WEB INTRODUCTORY EXAMPLE Linear Models in Economics and Engineering It was late summer in 949. Harvard

### Mathematics Notes for Class 12 chapter 3. Matrices

1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

### Lecture 10: Invertible matrices. Finding the inverse of a matrix

Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices

### The Inverse of a Square Matrix

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### Definition: Group A group is a set G together with a binary operation on G, satisfying the following axioms: a (b c) = (a b) c.

Algebraic Structures Abstract algebra is the study of algebraic structures. Such a structure consists of a set together with one or more binary operations, which are required to satisfy certain axioms.

### Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011