Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.


 Leona Pitts
 2 years ago
 Views:
Transcription
1 Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n matrices and r is a scalar then we can define matrix addition via a 11 a 1n b 11 b 1n A + B = + a m1 a mn b m1 b mn a 11 + b 11 a 1n + b 1n = a m1 + b m 1 a mn + b mn and scalar multiplication for matrices via ra = r a 11 a 1n ra 11 ra 1n = a m1 a mn ra m 1 ra mn. That is, adding and scalar multiplying is performed entrywise. It is important to note that two matrices must have the same size if we wish to add them together (and their sum has the same size as well).
2 As with vector addition and scalar multiplication, the same familiar arithmetical properties hold: Theorem If A, B, C are matrices of the same size and r and s are scalars, then 1. A + B = B + A; 2. ( A + B) + C = A + ( B + C); 3. A + 0 = 0 + A (where 0 represents the zero matrix, the matrix of the same size as A all of whose entries are zero); 4. r( A + B) = ra + rb and (r + s) A = ra + sa; 5. r( sa ) = (rs) A. // We can also extend the definition of the matrixvector product we developed to include a much wider class of matrices. The motivation comes from a consideration of how we compose linear transformations. As we have seen, any linear transformation S:R n R m has an associated standard m n matrix A for which S(y ) = Ay; here, y is an arbitrary vector in R n and its image S(y ) = Ay lies in R m. If we now consider a second linear transformation T:R p R n that has associated standard n p matrix B, then every vector x in R p is taken by T to the vector T(x ) = Bx in R n. Since
3 the output values of S can be used as input values for T, the composition map S T:R p R m makes sense: it carries a vector x in R p to the vector in R m. S T(x ) = S( T (x )) = A(Bx )) This composition map is a linear transformation (why?!), so it has a standard matrix representation. Therefore, we define the matrix product AB to be the standard matrix of the composition map S T. The first observation that must be made regarding this definition is that not every pair of matrices can be multiplied: AB only makes sense when the number of columns of A matches the number of rows of B. Next, we want to see how to determine the matrix product AB. By expressing the matrix B in terms of its columns as B = [ b 1 b 2 b p ], recall that Bx = x 1 b 1 + x 2 b x p b p is the linear combination of the columns of B using as weights the entries of x. Thus, A( Bx )) = A( x 1 b 1 + x 2 b x p b p ) = x 1 ( Ab 1 ) + x 2 ( Ab 2 ) + + x p ( Ab p )
4 is the linear combination of the matrixvector products of the columns of B by A using these same weights. This means that A( Bx )) = [ Ab 1 Ab 2 Ab p ]x, so that the matrix of the composition map must be given by the relation AB = [ Ab 1 Ab 2 Ab p ]. This shows exactly what the matrix product AB is. Because we know how to form the matrixvector products Ab j, we note that the columns of AB are the linear combinations of the columns of A whose weights come from the corresponding column of B. Equivalently, since the ith entry of the vector Ab j is the linear combination of the entries of the vector b j using as weights the entries of the ith row of A, we can conclude that the (i, j)entry of AB is given by the formula ( AB ) ij = a i 1 b 1 j + a i 2 b 2 j + + a in b nj.
5 This definition of matrix multiplication satisfies many of the familiar arithmetical properties we have come to expect (but it fails to satisfy some others!): Theorem Let A, B, C be matrices having appropriate sizes to allow the products below to be defined. Then 1. A( BC ) = ( AB )C; 2. A( B + C ) = AB + AC and ( A + B)C = AC + BC; 3. r( AB) = (ra )B = A(rB) for any scalar r; and 4. if A is m n, then I m A = A = AI n where I m and I n are the appropriately sized identity matrices. // Notice which properties do not appear in the above theorem: in general, matrices do not commute, that is, the relation AB = BA is not true in general, even if the matrix products on both sides of this equation make sense! (See Example 7, p. 114.) Next, if three matrices satisfy the relation AB = AC, it is not necessarily the case that B = C, that is, the cancellation law does not hold in general for matrix multiplication (see Exercise 10, p. 116). Finally, it may happen that AB = 0 without either A or B being equal to the zero matrix, e.g.,
6 = Another useful and common matrix operation is the matrix transpose. The transpose of the m n matrix A is the n m matrix A T, whose rows are the columns of A, or equivalently, whose columns are the rows of A. A simple check of properties shows that the transpose operation is compatible with matrix arithmetic: Theorem Let A, B be matrices having appropriate sizes to allow the expressions below to be defined. Then 1. ( A T ) T = A; 2. ( A + B) T = A T + B T ; 3. (ra ) T = ra T for any scalar r; and 4. ( AB ) T = B T A T. Proof Properties 13 are straightforward. To see why property 4 holds, we note that the entries in the ith row of (AB) T are the entries of the ith column of AB, which is Ab i ; the jth entry in this column is the linear combination of the entries of b i (the ith column of B) using weights taken from the jth row of A. That is, the (i, j)entry of (AB) T is the quantity a j1 b 1i +a j2 b 2i + +a jn b ni.
7 On the other hand, the (i, j)entry of B T A T is the linear combination of the entries in the jth column of A T using weights that come from the ith row of B T, which is equivalently the linear combination of the entries in the jth row of A using weights that come from the ith column of B. That is, the (i, j)entry of B T A T is the quantity b 1i a j1 +b 2i a j2 + +b ni a jn. Since this is identical to the computation in the previous paragraph, we must have that ( AB ) T = B T A T. //
Matrix Algebra 2.1 MATRIX OPERATIONS Pearson Education, Inc.
2 Matrix Algebra 2.1 MATRIX OPERATIONS MATRIX OPERATIONS m n If A is an matrixthat is, a matrix with m rows and n columnsthen the scalar entry in the ith row and jth column of A is denoted by a ij and
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x
More informationICS 6N Computational Linear Algebra Matrix Algebra
ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix
More informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationMath 4377/6308 Advanced Linear Algebra
2.3 Composition Math 4377/6308 Advanced Linear Algebra 2.3 Composition of Linear Transformations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377
More informationKevin James. MTHSC 3110 Section 2.1 Matrix Operations
MTHSC 3110 Section 2.1 Matrix Operations Notation Let A be an m n matrix, that is, m rows and n columns. We ll refer to the entries of A by their row and column indices. The entry in the i th row and j
More informationMathematics 13: Lecture 10
Mathematics 13: Lecture 10 Matrices Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, 2008 1 / 19 Matrices Recall: A matrix is a
More information10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )
c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.27.4) 1. When each of the functions F 1, F 2,..., F n in righthand side
More informationA FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION AttributionNonCommercialShareAlike (CC BYNCSA)
More informationMatrix 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 informationCS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:
CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n
More informationStagestructured Populations
Department of Biology New Mexico State University Las Cruces, New Mexico 88003 brook@nmsu.edu Fall 2009 AgeStructured Populations All individuals are not equivalent to each other Rates of survivorship
More informationMatrices: 2.1 Operations with Matrices
Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,
More informationMATRICES. a m,1 a m,n A =
MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of
More informationMatrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated
More informationa 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.
Chapter 1 LINEAR EQUATIONS 11 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 real
More information3. 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 informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29 Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationMath 3191 Applied Linear Algebra
Math 191 Applied Linear Algebra Lecture 8: Inverse of a Matrix Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/0 Announcements We will not make it to section. tonight,
More informationSection 9.2: Matrices.. a m1 a m2 a mn
Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in
More informationWe could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2
Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1
More informationMatrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation.
function Matrices Calculus III Summer 2013, Session II Wednesday, July 10, 2013 Agenda function 1. 2. function function Definition An m n matrix is a rectangular array of numbers arranged in m horizontal
More information1 Matrices and matrix algebra
1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns
More informationChapter 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 informationCLASS 12 ALGEBRA OF MATRICES
CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements
More informationMatrix Arithmetic. j=1
An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the ith row and jth column
More informationMath 360 Linear Algebra Fall Class Notes. a a a a a a. a a a
Math 360 Linear Algebra Fall 2008 91008 Class Notes Matrices As we have already seen, a matrix is a rectangular array of numbers. If a matrix A has m columns and n rows, we say that its dimensions are
More informationChapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
More informationSection 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.
Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A
More informationPrepared by: M. S. KumarSwamy, TGT(Maths) Page
Prepared by: M. S. KumarSwamy, TGT(Maths) Page  50  CHAPTER 3: MATRICES QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 03 marks Matrix A matrix is an ordered rectangular array of numbers
More informationFall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop
Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego RowColumn Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding
More informationLinear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Systems of Linear Equations Matrices Algebraic Properties of Matrix Operations Special Types of Matrices and Partitioned Matrices Matrix Transformations
More information[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of
. Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,
More informationElementary Row Operations on Matrices
King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix
More informationAnnouncements Monday, October 02
Announcements Monday, October 02 Please fill out the midsemester survey under Quizzes on Canvas WeBWorK 18, 19 are due Wednesday at 11:59pm The quiz on Friday covers 17, 18, and 19 My office is Skiles
More informationAnnouncements Wednesday, October 10
Announcements Wednesday, October 10 The second midterm is on Friday, October 19 That is one week from this Friday The exam covers 35, 36, 37, 39, 41, 42, 43, 44 (through today s material) WeBWorK 42, 43
More informationMatrix Algebra & Elementary Matrices
Matrix lgebra & Elementary Matrices To add two matrices, they must have identical dimensions. To multiply them the number of columns of the first must equal the number of rows of the second. The laws below
More informationPhys 201. Matrices and Determinants
Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1
More informationL. Vandenberghe EE133A (Spring 2017) 3. Matrices. notation and terminology. matrix operations. linear and affine functions.
L Vandenberghe EE133A (Spring 2017) 3 Matrices notation and terminology matrix operations linear and affine functions complexity 31 Matrix a rectangular array of numbers, for example A = 0 1 23 01 13
More informationMatrices BUSINESS MATHEMATICS
Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question
More informationKevin James. MTHSC 3110 Section 2.2 Inverses of Matrices
MTHSC 3110 Section 2.2 Inverses of Matrices Definition Suppose that T : R n R m is linear. We will say that T is invertible if for every b R m there is exactly one x R n so that T ( x) = b. Note If T is
More informationMatrix Multiplication
3.2 Matrix Algebra Matrix Multiplication Example Foxboro Stadium has three main concession stands, located behind the south, north and west stands. The topselling items are peanuts, hot dogs and soda.
More informationAppendix C Vector and matrix algebra
Appendix C Vector and matrix algebra Concepts Scalars Vectors, rows and columns, matrices Adding and subtracting vectors and matrices Multiplying them by scalars Products of vectors and matrices, scalar
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 20172018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2
More informationLecture 3 Linear Algebra Background
Lecture 3 Linear Algebra Background Dan Sheldon September 17, 2012 Motivation Preview of next class: y (1) w 0 + w 1 x (1) 1 + w 2 x (1) 2 +... + w d x (1) d y (2) w 0 + w 1 x (2) 1 + w 2 x (2) 2 +...
More informationMTH 35, SPRING 2017 NIKOS APOSTOLAKIS
MTH 35, SPRING 2017 NIKOS APOSTOLAKIS 1. Linear transformations Definition 1. A function T : R n R m is called a linear transformation if, for any scalars λ,µ R and any vectors u,v R n we have: T(λu+µv)
More informationExample. We can represent the information on July sales more simply as
CHAPTER 1 MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS 11 Matrices and Vectors In many occasions, we can arrange a number of values of interest into an rectangular array For example: Example We can
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationchapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS 5.1 Basic Definitions
chapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS The purpose of this chapter is to introduce you to matrix algebra, which has many applications. You are already familiar with several algebras: elementary
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
More informationTwo matrices of the same size are added by adding their corresponding entries =.
2 Matrix algebra 2.1 Addition and scalar multiplication Two matrices of the same size are added by adding their corresponding entries. For instance, 1 2 3 2 5 6 3 7 9 +. 4 0 9 4 1 3 0 1 6 Addition of two
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J Olver 3 Review of Matrix Algebra Vectors and matrices are essential for modern analysis of systems of equations algebrai, differential, functional, etc In this
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationMatrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix
Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix Matrix Operations Matrix Addition and Matrix Scalar Multiply Matrix Multiply Matrix
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 998 Comments to the author at krm@mathsuqeduau All contents copyright c 99 Keith
More informationA 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
More informationCalculus II  Basic Matrix Operations
Calculus II  Basic Matrix Operations Ryan C Daileda Terminology A matrix is a rectangular array of numbers, for example 7,, 7 7 9, or / / /4 / / /4 / / /4 / /6 The numbers in any matrix are called its
More informationMatrixMatrix Multiplication
Chapter MatrixMatrix Multiplication In this chapter, we discuss matrixmatrix multiplication We start by motivating its definition Next, we discuss why its implementation inherently allows high performance
More informationMatrix Algebra: Definitions and Basic Operations
Section 4 Matrix Algebra: Definitions and Basic Operations Definitions Analyzing economic models often involve working with large sets of linear equations. Matrix algebra provides a set of tools for dealing
More information7.5 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 informationLinear Algebra Summary. Based on Linear Algebra and its applications by David C. Lay
Linear Algebra Summary Based on Linear Algebra and its applications by David C. Lay Preface The goal of this summary is to offer a complete overview of all theorems and definitions introduced in the chapters
More information2.1 Matrices. 3 5 Solve for the variables in the following matrix equation.
2.1 Matrices Reminder: A matrix with m rows and n columns has size m x n. (This is also sometimes referred to as the order of the matrix.) The entry in the ith row and jth column of a matrix A is denoted
More information1300 Linear Algebra and Vector Geometry
1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca MayJune 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More informationMatrices. Chapter Definitions and Notations
Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which
More informationTopic 1: Matrix diagonalization
Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it
More informationMATH 2030: MATRICES ,, a m1 a m2 a mn If the columns of A are the vectors a 1, a 2,...,a n ; A is represented as A 1. .
MATH 030: MATRICES Matrix Operations We have seen how matrices and the operations on them originated from our study of linear equations In this chapter we study matrices explicitely Definition 01 A matrix
More informationFinite 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 informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationChapter 4  MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4  MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationJim Lambers MAT 610 Summer Session Lecture 1 Notes
Jim Lambers MAT 60 Summer Session 20090 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra
More informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces PerOlof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field
More informationLinear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds
Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds These notes are meant to provide a brief introduction to the topics from Linear Algebra that will be useful in Math3315/CSE3365, Introduction
More informationis 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 informationIntroduction. Vectors and Matrices. Vectors [1] Vectors [2]
Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector  one dimensional array Matrix 
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 6 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 6 1 / 14 Gauss Jordan elimination Last time we discussed bringing matrices to reduced
More informationChapter 2. Ma 322 Fall Ma 322. Sept 2327
Chapter 2 Ma 322 Fall 2013 Ma 322 Sept 2327 Summary ˆ Matrices and their Operations. ˆ Special matrices: Zero, Square, Identity. ˆ Elementary Matrices, Permutation Matrices. ˆ Voodoo Principle. What is
More informationThink about systems of linear equations, their solutions, and how they might be represented with matrices.
Think About This Situation Unit 4 Lesson 3 Investigation 1 Name: Think about systems of linear equations, their solutions, and how they might be represented with matrices. a Consider a system of two linear
More informationn n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full
n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x
More informationINSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES
1 CHAPTER 4 MATRICES 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES 1 Matrices Matrices are of fundamental importance in 2dimensional and 3dimensional graphics programming
More informationIntroduction to Matrices and Linear Systems Ch. 3
Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts
More informationReview Let A, B, and C be matrices of the same size, and let r and s be scalars. Then
1 Sec 21 Matrix Operations Review Let A, B, and C be matrices of the same size, and let r and s be scalars Then (i) A + B = B + A (iv) r(a + B) = ra + rb (ii) (A + B) + C = A + (B + C) (v) (r + s)a = ra
More informationMatrix representation of a linear map
Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors
More information. a m1 a mn. a 1 a 2 a = a n
Biostat 140655, 2008: Matrix Algebra Review 1 Definition: An m n matrix, A m n, is a rectangular array of real numbers with m rows and n columns Element in the i th row and the j th column is denoted by
More informationDETERMINANTS. , x 2 = a 11b 2 a 21 b 1
DETERMINANTS 1 Solving linear equations The simplest type of equations are linear The equation (1) ax = b is a linear equation, in the sense that the function f(x) = ax is linear 1 and it is equated to
More informationChapter 4. Vector Space Examples. 4.1 Diffusion Welding and Heat States
Chapter 4 Vector Space Examples 4.1 Diffusion Welding and Heat States In this section, we begin a deeper look into the mathematics for diffusion welding application discussed in Chapter 1. Recall that
More informationLecture 3: Matrix and Matrix Operations
Lecture 3: Matrix and Matrix Operations Representation, row vector, column vector, element of a matrix. Examples of matrix representations Tables and spreadsheets ScalarMatrix operation: Scaling a matrix
More informationDM559 Linear and Integer Programming. Lecture 3 Matrix Operations. Marco Chiarandini
DM559 Linear and Integer Programming Lecture 3 Matrix Operations Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline and 1 2 3 and 4 2 Outline and 1 2
More information3 Matrix Algebra. 3.1 Operations on matrices
3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8
More informationChapter 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 informationMAT 2037 LINEAR ALGEBRA I web:
MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut http://kisideuedutr/enginmermut/ HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear
More informationMath 220 F11 Lecture Notes
Math 22 F Lecture Notes William Chen November 4, 2. Lecture. Firstly, lets just get some notation out of the way. Notation. R, Q, C, Z, N,,,, {},, A B. Everyone in high school should have studied equations
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationChapter 2: Matrices and Linear Systems
Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers
More informationMAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :
MAC 0 Module Systems of Linear Equations and Matrices II Learning Objectives Upon completing this module, you should be able to :. Find the inverse of a square matrix.. Determine whether a matrix is invertible..
More information