System of Linear Equations

Size: px
Start display at page:

Download "System of Linear Equations"

Transcription

1 Chapter 7 - S&B

2 Gaussian and Gauss-Jordan Elimination We will study systems of linear equations by describing techniques for solving such systems. The preferred solution technique- Gaussian elimination- answers the fundamental questions about a given linear system: does a solution exist, and if so, how many solutions are there? In general 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 2.. =. a m1 x 1 + a m2 x a mn x n = b m In this system, the a ij s and b i s are given real numbers; a ij is the coeffi cient of the unknown x j in the ith equation. A solution of a system is an n-tuple of real numbers x 1, x 2,.,x n, which satisfies each of the m equations.

3 Gaussian and Gauss-Jordan Elimination We are interested in the following three questions: 1 Does a solution exist? 2 How many solutions are there? 3 Is there an effi cient algorithm that computes actual solutions? There are three ways of solving such systems: (1) Substitution, (2) elimination of variables, and (3) matrix methods.

4 Substitution Solve one equation of system for one variable, say x n in terms of the other variables in that equation. Substitute this expression for x n into the other m 1 equations. The result is a new system of m 1 equations in the n 1unknowns x 1,..., x n 1 Continue this process by solving one equation in the new system for x n 1 and substituting this expression into the other m 2 equations to obtain a system of m 2 equations in the n 2 variables x 1,..., x n 2.

5 Substitution x 1 = 0.4x x x 2 = 0.2x x x x 3 = 0.5x x x

6 Elimination of variables x 1 0.4x 2 0.3x 3 = x x x 3 = x x x 3 = 95 We will also have to use the method back substitution. This procedure is called Gaussian elimination.

7 Elementary Row Operations Coeffi cient Matrix and Augmented Matrix Example x 1 2x 2 = 8 3x 1 + x 2 = 3 ( ) 1 2 and 3 1 ( ) Our three elementary equation operations now become elementary row operations: Interchange two rows of a matrix. Change a row by adding to it a multiple of another row, and Multiply each element in a row by the same non-zero number.

8 Elementary Row Operations Definition. A row of a matrix is said to have k leading zeros if the first k elements of the row are all zeros and the (k + 1)th element of the row is not zero. With this terminology, a matrix is in row echelon form if each row has more leading zeros than the row preceding it. Example ( ) and ( ) ( ) ( and )

9 Elementary Row Operations Identity Matrix I = Zero Matrix 0 =

10 Elementary Row Operations Pivot. It is the first nonzero entry in each row of a matrix in row echelon form Let us analyze the following example Definition. A row echelon matrix in which each pivot is a I and in which each column containing a pivot contains no other nonzero entries is said to be in reduced row echelon form.

11 System with many or no solutions In general two lines in the plane will be nonparallel and will cross in exactly one point. However, they can be parallel to each other. In this case, they will either coincide or they will never cross. lf they coincide; every point on either line is a solution (infinite many solutions). Let us analyze the following example p 1 + 2p 2 + 3p 3 = 1 3p 1 + 2p 2 + p 3 = 1

12 Rank - The Fundamental Criterion In order to answer the questions related to existence and uniqueness of solutions we need to analyze the rank. Definition.The rank of a matrix is the number of nonzero rows in its row echelon form. ( ) Fact 7.1. Let A and  be the coeffi cient matrix and augmented matrix respectively of a system of linear equations. Then rank A rank  rank A number of rows of A rank A number of columns of A

13 Fact 7.2. A system of linear equations with coeffi cient matrix A and augmented matrix  has a solution if and only if rank  = rank A Fact 7.3. A linear system of equations must have either no solutions, one solution, or infinitely many solutions. Fact 7.4. A system with a unique solution must have at least as many equations as unknowns. Homogeneous system (they have at least one solution) a 11 x a 1n x n = 0 a 21 x a 2n x n = 0... =. a m1 x a mn x n = 0

14 Fact 7.6. A homogeneous system of linear equations which has more unknows than equations must have infinitely many distinct solutions Fact 7.7. A system of linear equations with coeffi cient matrix A will have a solution for every choice of RHS b 1,.., b n if and only if rank A = number of rows of A Fact 7.9. Any system of linear equations having A as its coeffi cient matrix will have at most one solution for every choice of RHS b1,.., bm if and only if rank A = number of columns of A Fact A coeffi cient matrix A is nonsingular, that is, the corresponding linear system has one and only solution for every choice of RHS is number of rows of A =number of columns of A = rank A

15 Rank - The Fundamental Criterion Fact Consider the linear system of equations Ax = b If the number of equations < the number of unknowns, then: Ax = 0 has infinitely many solutions, for any given b, Ax = b has 0 or infinitely many solutions, and if rank A = number of equations, Ax = b has infinitely many solutions for every RHS b. If the number of equations = the number of unknowns, then: Ax = 0 has one or infinitely many solutions. for any given b, Ax = b has 0, 1, or infinitely many solutions, and if rank A = number of unknowns = number of equations, Ax = b has exactly 1 solution for every RHS b.

Chapter 1. Vectors, Matrices, and Linear Spaces

Chapter 1. Vectors, Matrices, and Linear Spaces 1.4 Solving Systems of Linear Equations 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.4. Solving Systems of Linear Equations Note. We give an algorithm for solving a system of linear equations (called

More information

Section Gaussian Elimination

Section Gaussian Elimination Section. - Gaussian Elimination A matrix is said to be in row echelon form (REF) if it has the following properties:. The first nonzero entry in any row is a. We call this a leading one or pivot one..

More information

1 - Systems of Linear Equations

1 - Systems of Linear Equations 1 - Systems of Linear Equations 1.1 Introduction to Systems of Linear Equations Almost every problem in linear algebra will involve solving a system of equations. ü LINEAR EQUATIONS IN n VARIABLES We are

More information

Math 1314 Week #14 Notes

Math 1314 Week #14 Notes Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,

More information

Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination

Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University September 22, 2017 Review: The coefficient matrix Consider a system of m linear equations in n variables.

More information

Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) 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 and b are

More information

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra A linear (algebraic) 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 and b are real numbers. 1

More information

Chapter 2. Systems of Equations and Augmented Matrices. Creighton University

Chapter 2. Systems of Equations and Augmented Matrices. Creighton University Chapter Section - Systems of Equations and Augmented Matrices D.S. Malik Creighton University Systems of Linear Equations Common ways to solve a system of equations: Eliminationi Substitution Elimination

More information

Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) 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 and b are

More information

Problem Sheet 1 with Solutions GRA 6035 Mathematics

Problem Sheet 1 with Solutions GRA 6035 Mathematics Problem Sheet 1 with Solutions GRA 6035 Mathematics BI Norwegian Business School 2 Problems 1. From linear system to augmented matrix Write down the coefficient matrix and the augmented matrix of the following

More information

Solving Linear Systems Using Gaussian Elimination

Solving Linear Systems Using Gaussian Elimination Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n

More information

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Midterm 1 Review Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Summary This Midterm Review contains notes on sections 1.1 1.5 and 1.7 in your

More information

Linear Algebra I Lecture 8

Linear Algebra I Lecture 8 Linear Algebra I Lecture 8 Xi Chen 1 1 University of Alberta January 25, 2019 Outline 1 2 Gauss-Jordan Elimination Given a system of linear equations f 1 (x 1, x 2,..., x n ) = 0 f 2 (x 1, x 2,..., x n

More information

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix

More information

Notes on Row Reduction

Notes on Row Reduction Notes on Row Reduction Francis J. Narcowich Department of Mathematics Texas A&M University September The Row-Reduction Algorithm The row-reduced form of a matrix contains a great deal of information, both

More information

Lectures on Linear Algebra for IT

Lectures on Linear Algebra for IT Lectures on Linear Algebra for IT by Mgr. Tereza Kovářová, Ph.D. following content of lectures by Ing. Petr Beremlijski, Ph.D. Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 2. Systems

More information

1 Last time: linear systems and row operations

1 Last time: linear systems and row operations 1 Last time: linear systems and row operations Here s what we did last time: a system of linear equations or linear system is a list of equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22

More information

Section 6.2 Larger Systems of Linear Equations

Section 6.2 Larger Systems of Linear Equations Section 6.2 Larger Systems of Linear Equations Gaussian Elimination In general, to solve a system of linear equations using its augmented matrix, we use elementary row operations to arrive at a matrix

More information

Linear Algebra I Lecture 10

Linear Algebra I Lecture 10 Linear Algebra I Lecture 10 Xi Chen 1 1 University of Alberta January 30, 2019 Outline 1 Gauss-Jordan Algorithm ] Let A = [a ij m n be an m n matrix. To reduce A to a reduced row echelon form using elementary

More information

1.4 Gaussian Elimination Gaussian elimination: an algorithm for finding a (actually the ) reduced row echelon form of a matrix. A row echelon form

1.4 Gaussian Elimination Gaussian elimination: an algorithm for finding a (actually the ) reduced row echelon form of a matrix. A row echelon form 1. Gaussian Elimination Gaussian elimination: an algorithm for finding a (actually the ) reduced row echelon form of a matrix. Original augmented matrix A row echelon form 1 1 0 0 0 1!!!! The reduced row

More information

DM559 Linear and Integer Programming. Lecture 2 Systems of Linear Equations. Marco Chiarandini

DM559 Linear and Integer Programming. Lecture 2 Systems of Linear Equations. Marco Chiarandini DM559 Linear and Integer Programming Lecture Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. Outline 1. 3 A Motivating Example You are organizing

More information

Finite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.

Finite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero. Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form

More information

Gauss-Jordan Row Reduction and Reduced Row Echelon Form

Gauss-Jordan Row Reduction and Reduced Row Echelon Form Gauss-Jordan Row Reduction and Reduced Row Echelon Form If we put the augmented matrix of a linear system in reduced row-echelon form, then we don t need to back-substitute to solve the system. To put

More information

Matrices and RRE Form

Matrices and RRE Form Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that

More information

Matrix & Linear Algebra

Matrix & Linear Algebra Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A

More information

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,

More information

Chapter 3. Linear Equations. Josef Leydold Mathematical Methods WS 2018/19 3 Linear Equations 1 / 33

Chapter 3. Linear Equations. Josef Leydold Mathematical Methods WS 2018/19 3 Linear Equations 1 / 33 Chapter 3 Linear Equations Josef Leydold Mathematical Methods WS 2018/19 3 Linear Equations 1 / 33 Lineares Gleichungssystem System of m linear equations in n unknowns: a 11 x 1 + a 12 x 2 + + a 1n x n

More information

Linear System Equations

Linear System Equations King Saud University September 24, 2018 Table of contents 1 2 3 4 Definition A linear system of equations with m equations and n unknowns is defined as follows: a 1,1 x 1 + a 1,2 x 2 + + a 1,n x n = b

More information

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2

More information

MAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices

MAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices MAC05-College Algebra Chapter 5-Systems of Equations & Matrices 5. Systems of Equations in Two Variables Solving Systems of Two Linear Equations/ Two-Variable Linear Equations A system of equations is

More information

Chapter 1: Systems of Linear Equations

Chapter 1: Systems of Linear Equations Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture 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

More information

3. Replace any row by the sum of that row and a constant multiple of any other row.

3. Replace any row by the sum of that row and a constant multiple of any other row. Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or

More information

Methods for Solving Linear Systems Part 2

Methods for Solving Linear Systems Part 2 Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use

More information

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number

More information

Matrices and systems of linear equations

Matrices and systems of linear equations Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION x 1,, x n A linear equation in the variables equation that can be written in the form a 1 x 1 + a 2 x 2 + + a n x n

More information

Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra

Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent

More information

Chapter 4. Solving Systems of Equations. Chapter 4

Chapter 4. Solving Systems of Equations. Chapter 4 Solving Systems of Equations 3 Scenarios for Solutions There are three general situations we may find ourselves in when attempting to solve systems of equations: 1 The system could have one unique solution.

More information

MTH 2530: Linear Algebra. Sec Systems of Linear Equations

MTH 2530: Linear Algebra. Sec Systems of Linear Equations MTH 0 Linear Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Week # Section.,. Sec... Systems of Linear Equations... D examples Example Consider a system of

More information

MATH 2050 Assignment 6 Fall 2018 Due: Thursday, November 1. x + y + 2z = 2 x + y + z = c 4x + 2z = 2

MATH 2050 Assignment 6 Fall 2018 Due: Thursday, November 1. x + y + 2z = 2 x + y + z = c 4x + 2z = 2 MATH 5 Assignment 6 Fall 8 Due: Thursday, November [5]. For what value of c does have a solution? Is it unique? x + y + z = x + y + z = c 4x + z = Writing the system as an augmented matrix, we have c R

More information

Section 1.1: Systems of Linear Equations

Section 1.1: Systems of Linear Equations Section 1.1: Systems of Linear Equations Two Linear Equations in Two Unknowns Recall that the equation of a line in 2D can be written in standard form: a 1 x 1 + a 2 x 2 = b. Definition. A 2 2 system of

More information

Math "Matrix Approach to Solving Systems" Bibiana Lopez. November Crafton Hills College. (CHC) 6.3 November / 25

Math Matrix Approach to Solving Systems Bibiana Lopez. November Crafton Hills College. (CHC) 6.3 November / 25 Math 102 6.3 "Matrix Approach to Solving Systems" Bibiana Lopez Crafton Hills College November 2010 (CHC) 6.3 November 2010 1 / 25 Objectives: * Define a matrix and determine its order. * Write the augmented

More information

EBG # 3 Using Gaussian Elimination (Echelon Form) Gaussian Elimination: 0s below the main diagonal

EBG # 3 Using Gaussian Elimination (Echelon Form) Gaussian Elimination: 0s below the main diagonal EBG # 3 Using Gaussian Elimination (Echelon Form) Gaussian Elimination: 0s below the main diagonal [ x y Augmented matrix: 1 1 17 4 2 48 (Replacement) Replace a row by the sum of itself and a multiple

More information

MODEL ANSWERS TO THE THIRD HOMEWORK

MODEL ANSWERS TO THE THIRD HOMEWORK MODEL ANSWERS TO THE THIRD HOMEWORK 1 (i) We apply Gaussian elimination to A First note that the second row is a multiple of the first row So we need to swap the second and third rows 1 3 2 1 2 6 5 7 3

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2 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

More information

Exercise Sketch these lines and find their intersection.

Exercise Sketch these lines and find their intersection. These are brief notes for the lecture on Friday August 21, 2009: they are not complete, but they are a guide to what I want to say today. They are not guaranteed to be correct. 1. Solving systems of linear

More information

Relationships Between Planes

Relationships Between Planes Relationships Between Planes Definition: consistent (system of equations) A system of equations is consistent if there exists one (or more than one) solution that satisfies the system. System 1: {, System

More information

4 Elementary matrices, continued

4 Elementary matrices, continued 4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices are constructed

More information

Lecture 22: Section 4.7

Lecture 22: Section 4.7 Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n

More information

10. Rank-nullity Definition Let A M m,n (F ). The row space of A is the span of the rows. The column space of A is the span of the columns.

10. Rank-nullity Definition Let A M m,n (F ). The row space of A is the span of the rows. The column space of A is the span of the columns. 10. Rank-nullity Definition 10.1. Let A M m,n (F ). The row space of A is the span of the rows. The column space of A is the span of the columns. The nullity ν(a) of A is the dimension of the kernel. The

More information

4 Elementary matrices, continued

4 Elementary matrices, continued 4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row

More information

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.

More information

MAC Module 1 Systems of Linear Equations and Matrices I

MAC Module 1 Systems of Linear Equations and Matrices I MAC 2103 Module 1 Systems of Linear Equations and Matrices I 1 Learning Objectives Upon completing this module, you should be able to: 1. Represent a system of linear equations as an augmented matrix.

More information

Linear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.

Linear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations. POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems

More information

Linear Independence x

Linear Independence x Linear Independence A consistent system of linear equations with matrix equation Ax = b, where A is an m n matrix, has a solution set whose graph in R n is a linear object, that is, has one of only n +

More information

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding Elementary matrices, continued To summarize, we have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices

More information

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3 ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3 ISSUED 24 FEBRUARY 2018 1 Gaussian elimination Let A be an (m n)-matrix Consider the following row operations on A (1) Swap the positions any

More information

1 Determinants. 1.1 Determinant

1 Determinants. 1.1 Determinant 1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to

More information

7.6 The Inverse of a Square Matrix

7.6 The Inverse of a Square Matrix 7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses

More information

Chapter 1 Matrices and Systems of Equations

Chapter 1 Matrices and Systems of Equations Chapter 1 Matrices and Systems of Equations System of Linear Equations 1. A linear equation in n unknowns is an equation of the form n i=1 a i x i = b where a 1,..., a n, b R and x 1,..., x n are variables.

More information

Row Reduction and Echelon Forms

Row Reduction and Echelon Forms Row Reduction and Echelon Forms 1 / 29 Key Concepts row echelon form, reduced row echelon form pivot position, pivot, pivot column basic variable, free variable general solution, parametric solution existence

More information

Lecture Notes: Solving Linear Systems with Gauss Elimination

Lecture Notes: Solving Linear Systems with Gauss Elimination Lecture Notes: Solving Linear Systems with Gauss Elimination Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Echelon Form and Elementary

More information

The matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0.

The matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0. ) Find all solutions of the linear system. Express the answer in vector form. x + 2x + x + x 5 = 2 2x 2 + 2x + 2x + x 5 = 8 x + 2x + x + 9x 5 = 2 2 Solution: Reduce the augmented matrix [ 2 2 2 8 ] to

More information

POLI270 - Linear Algebra

POLI270 - Linear Algebra POLI7 - Linear Algebra Septemer 8th Basics a x + a x +... + a n x n b () is the linear form where a, b are parameters and x n are variables. For a given equation such as x +x you only need a variable and

More information

SOLVING Ax = b: GAUSS-JORDAN ELIMINATION [LARSON 1.2]

SOLVING Ax = b: GAUSS-JORDAN ELIMINATION [LARSON 1.2] SOLVING Ax = b: GAUSS-JORDAN ELIMINATION [LARSON.2 EQUIVALENT LINEAR SYSTEMS: Two m n linear systems are equivalent both systems have the exact same solution sets. When solving a linear system Ax = b,

More information

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ] Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.

More information

Solving Systems of Linear Equations

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

More information

7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved.

7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved. 7.5 Operations with Matrices Copyright Cengage Learning. All rights reserved. What You Should Learn Decide whether two matrices are equal. Add and subtract matrices and multiply matrices by scalars. Multiply

More information

ORIE 6300 Mathematical Programming I August 25, Recitation 1

ORIE 6300 Mathematical Programming I August 25, Recitation 1 ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Calvin Wylie Recitation 1 Scribe: Mateo Díaz 1 Linear Algebra Review 1 1.1 Independence, Spanning, and Dimension Definition 1 A (usually infinite)

More information

Matrix Solutions to Linear Equations

Matrix Solutions to Linear Equations Matrix Solutions to Linear Equations Augmented matrices can be used as a simplified way of writing a system of linear equations. In an augmented matrix, a vertical line is placed inside the matrix to represent

More information

5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns

5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns 5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns (1) possesses the solution and provided that.. The numerators and denominators are recognized

More information

Linear equations in linear algebra

Linear equations in linear algebra Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear

More information

LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS

LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1 Linear equations We now switch gears to discuss the topic of solving linear equations, and more interestingly, systems

More information

1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic

1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic 1300 Linear Algebra and Vector Geometry Week 2: Jan 14 18 1.2, 1.3... Gauss-Jordan, homogeneous matrices, intro matrix arithmetic R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca Winter 2019 What

More information

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra A linear (algebraic) 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 and b are real numbers. 1

More information

Lecture 3: Gaussian Elimination, continued. Lecture 3: Gaussian Elimination, continued

Lecture 3: Gaussian Elimination, continued. Lecture 3: Gaussian Elimination, continued Definition The process of solving a system of linear equations by converting the system to an augmented matrix is called Gaussian Elimination. The general strategy is as follows: Convert the system of

More information

MathQuest: Linear Algebra. 1. Which of the following operations on an augmented matrix could change the solution set of a system?

MathQuest: Linear Algebra. 1. Which of the following operations on an augmented matrix could change the solution set of a system? MathQuest: Linear Algebra Gaussian Elimination 1. Which of the following operations on an augmented matrix could change the solution set of a system? (a) Interchanging two rows (b) Multiplying one row

More information

Math 313 Chapter 1 Review

Math 313 Chapter 1 Review Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations

More information

9.1 - Systems of Linear Equations: Two Variables

9.1 - Systems of Linear Equations: Two Variables 9.1 - Systems of Linear Equations: Two Variables Recall that a system of equations consists of two or more equations each with two or more variables. A solution to a system in two variables is an ordered

More information

Linear Algebra. James Je Heon Kim

Linear Algebra. James Je Heon Kim Linear lgebra James Je Heon Kim (jjk9columbia.edu) If you are unfamiliar with linear or matrix algebra, you will nd that it is very di erent from basic algebra or calculus. For the duration of this session,

More information

March 19 - Solving Linear Systems

March 19 - Solving Linear Systems March 19 - Solving Linear Systems Welcome to linear algebra! Linear algebra is the study of vectors, vector spaces, and maps between vector spaces. It has applications across data analysis, computer graphics,

More information

Extra Problems: Chapter 1

Extra Problems: Chapter 1 MA131 (Section 750002): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 1 Extra Problems: Chapter 1 1. In each of the following answer true if the statement is always true and false otherwise in the space

More information

3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions

3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence

More information

Lecture 2 Systems of Linear Equations and Matrices, Continued

Lecture 2 Systems of Linear Equations and Matrices, Continued Lecture 2 Systems of Linear Equations and Matrices, Continued Math 19620 Outline of Lecture Algorithm for putting a matrix in row reduced echelon form - i.e. Gauss-Jordan Elimination Number of Solutions

More information

3.4 Elementary Matrices and Matrix Inverse

3.4 Elementary Matrices and Matrix Inverse Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary

More information

Graduate Mathematical Economics Lecture 1

Graduate Mathematical Economics Lecture 1 Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists

More information

Undergraduate Mathematical Economics Lecture 1

Undergraduate Mathematical Economics Lecture 1 Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where

More information

Linear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey

Linear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey Copyright 2005, W.R. Winfrey Topics Preliminaries Echelon Form of a Matrix Elementary Matrices; Finding A -1 Equivalent Matrices LU-Factorization Topics Preliminaries Echelon Form of a Matrix Elementary

More information

Math 314 Lecture Notes Section 006 Fall 2006

Math 314 Lecture Notes Section 006 Fall 2006 Math 314 Lecture Notes Section 006 Fall 2006 CHAPTER 1 Linear Systems of Equations First Day: (1) Welcome (2) Pass out information sheets (3) Take roll (4) Open up home page and have students do same

More information

Determine whether the following system has a trivial solution or non-trivial solution:

Determine whether the following system has a trivial solution or non-trivial solution: Practice Questions Lecture # 7 and 8 Question # Determine whether the following system has a trivial solution or non-trivial solution: x x + x x x x x The coefficient matrix is / R, R R R+ R The corresponding

More information

13. Systems of Linear Equations 1

13. Systems of Linear Equations 1 13. Systems of Linear Equations 1 Systems of linear equations One of the primary goals of a first course in linear algebra is to impress upon the student how powerful matrix methods are in solving systems

More information

Solutions to Homework 5 - Math 3410

Solutions to Homework 5 - Math 3410 Solutions to Homework 5 - Math 34 (Page 57: # 489) Determine whether the following vectors in R 4 are linearly dependent or independent: (a) (, 2, 3, ), (3, 7,, 2), (, 3, 7, 4) Solution From x(, 2, 3,

More information

Department of Aerospace Engineering AE602 Mathematics for Aerospace Engineers Assignment No. 4

Department of Aerospace Engineering AE602 Mathematics for Aerospace Engineers Assignment No. 4 Department of Aerospace Engineering AE6 Mathematics for Aerospace Engineers Assignment No.. Decide whether or not the following vectors are linearly independent, by solving c v + c v + c 3 v 3 + c v :

More information

Pre-Calculus I. For example, the system. x y 2 z. may be represented by the augmented matrix

Pre-Calculus I. For example, the system. x y 2 z. may be represented by the augmented matrix Pre-Calculus I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural

More information

MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics

MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones

More information

Section 5.3 Systems of Linear Equations: Determinants

Section 5.3 Systems of Linear Equations: Determinants Section 5. Systems of Linear Equations: Determinants In this section, we will explore another technique for solving systems called Cramer's Rule. Cramer's rule can only be used if the number of equations

More information

Lecture 7: Introduction to linear systems

Lecture 7: Introduction to linear systems Lecture 7: Introduction to linear systems Two pictures of linear systems Consider the following system of linear algebraic equations { x 2y =, 2x+y = 7. (.) Note that it is a linear system with two unknowns

More information

Lecture 1 Systems of Linear Equations and Matrices

Lecture 1 Systems of Linear Equations and Matrices Lecture 1 Systems of Linear Equations and Matrices Math 19620 Outline of Course Linear Equations and Matrices Linear Transformations, Inverses Bases, Linear Independence, Subspaces Abstract Vector Spaces

More information

1. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. The augmented matrix of this linear system is

1. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. The augmented matrix of this linear system is Solutions to Homework Additional Problems. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. (a) x + y = 8 3x + 4y = 7 x + y = 3 The augmented matrix of this linear system

More information