Matrix Basic Concepts

Size: px
Start display at page:

Download "Matrix Basic Concepts"

Transcription

1 Matrix Basic Concepts Topics: What is a matrix? Matrix terminology Elements or entries Diagonal entries Address/location of entries Rows and columns Size of a matrix A column matrix; vectors Special types of matrices Square Diagonal Triangular Identity matrix Operations on matrices Equal matrices Addition, subtraction A number (scalar) times a matrix A matrix times a column matrix

2 Common Matrix Terminology B 10 A matrix is a rectangular array of rows and columns. For example. We usually name matrices using letters; for instance, letters are often used.). (Capital This convention lets us easily refer to a matrix using the letter assigned, here B. The numerical values within a matrix are called elements or entries of the matrix. Each matrix entry has an address/location given by the number of the row and the number of the column in which it appears. Entries of a matrix are often denoted by a lower case letter of it s assigned name with a pair of subscripts denoting the row, column position of the entry. 4 B 10 Example: For we have b 11 = 4, b 13 = π, and b 1 = 7. The entries occupying like numbered row, column position are referred to as diagonal entries. For the matrix B in the preceding example the diagonal entries are b 11 = 4 and b = 3.

3 4 B 10 The size of a matrix is denoted by listing the number of rows followed by the number of columns. Matrix B is 3; it has two rows and 3 columns. The number of rows is always stated first. A column matrix consists of a number of rows and a single column. Each of the following is a column matrix. It is common practice to use lower case letters for column matrices. We have indicated the size of the column below the matrix. w 1 1 x1 w 0 x w 3 w, z, x Column matrices are often called vectors. This is from the geometric notions of directed line segments in the plane or 3-space. Such geometric vectors are designated by the coordinates of the end point of the line segment when we draw it starting from the origin. 4

4 Special types of matrices. A square matrix has the same number of rows as column. A diagonal matrix is a square matrix in which the non-diagonal entries are all zero e A diagonal matrix with all diagonal entries equal to 1 is called an identity matrix. An upper triangular matrix is a square matrix with all zero entries below the diagonal etc Called zero matrices. Denoted I and I 3 respectively. The n n identity is denoted I n An lower triangular matrix is a square matrix with all zero entries above the diagonal

5 Matrix Operations Definition Two m n matrices A = [a ij ] and B = [b ij ] are said to be equal matrices if a ij = b ij for 1 i m, 1 j n, that is, if corresponding entries are equal. Definition If A = [a ij ] and B = [b ij ] are both m n matrices, then their sum, A + B, is the matrix whose (i,j)-entry is a ij + b ij ; that is, we add corresponding entries. The difference, A - B, is the matrix whose (i,j)-entry is a ij - b ij ; that is, we subtract corresponding entries A , and 6 8 B 0 1 A +B 6 7 A - B 6 9 The next operation involves numbers (possibly real or complex), which we call scalars and a matrix. The entries of the matrix could be real numbers, complex numbers, or even functions; we have not restricted the entries of a matrix. Definition If A = [a ij ] is an m n matrix and k is a scalar, then the scalar multiple of A by k, ka, is the m n matrix whose entries are ka ij ; that is, each entry of matrix A is multiplied by k. 4 1i 6 8i i 1i , i 0 i 0 4i i 6i i 4i 6

6 The matrix operations developed so far involved element-by-element manipulations. Here we introduce an operation that involves row-by-column dot products. Definition Let A be an m n matrix and c be a column matrix with n entries, that is, an n 1 matrix. Then the matrix A times column c is the n 1 matrix whose entries are obtained by taking dot product of each row of A with the column matrix c A 3 1 c Ac Warning: If the number of columns of A is not equal to the number of rows of c, then the product of A times c is NOT defined A c Ac

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of

[ 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 information

Matrices: 2.1 Operations with Matrices

Matrices: 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 information

Calculus II - Basic Matrix Operations

Calculus 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 information

Matrices and Vectors

Matrices and Vectors Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix

More information

Linear Algebra V = T = ( 4 3 ).

Linear Algebra V = T = ( 4 3 ). Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional

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

Matrix Algebra: Definitions and Basic Operations

Matrix 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 information

The Matrix Vector Product and the Matrix Product

The Matrix Vector Product and the Matrix Product The Matrix Vector Product and the Matrix Product As we have seen a matrix is just a rectangular array of scalars (real numbers) The size of a matrix indicates its number of rows and columns A matrix with

More information

Chapter 2. Ma 322 Fall Ma 322. Sept 23-27

Chapter 2. Ma 322 Fall Ma 322. Sept 23-27 Chapter 2 Ma 322 Fall 2013 Ma 322 Sept 23-27 Summary ˆ Matrices and their Operations. ˆ Special matrices: Zero, Square, Identity. ˆ Elementary Matrices, Permutation Matrices. ˆ Voodoo Principle. What is

More information

Finite Math - J-term Section Systems of Linear Equations in Two Variables Example 1. Solve the system

Finite Math - J-term Section Systems of Linear Equations in Two Variables Example 1. Solve the system Finite Math - J-term 07 Lecture Notes - //07 Homework Section 4. - 9, 0, 5, 6, 9, 0,, 4, 6, 0, 50, 5, 54, 55, 56, 6, 65 Section 4. - Systems of Linear Equations in Two Variables Example. Solve the system

More information

Matrices. Chapter Definitions and Notations

Matrices. 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 information

Definition. A matrix is a rectangular array of numbers enclosed by brackets (plural: matrices).

Definition. A matrix is a rectangular array of numbers enclosed by brackets (plural: matrices). Matrices (general theory). Definition. A matrix is a rectangular array of numbers enclosed by brackets (plural: matrices). Examples. 1 2 1 1 0 2 A= 0 0 7 B= 0 1 3 4 5 0 Terminology and Notations. Each

More information

Phys 201. Matrices and Determinants

Phys 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 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

Review of Linear Algebra

Review 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 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

4-1 Matrices and Data

4-1 Matrices and Data 4-1 Matrices and Data Warm Up Lesson Presentation Lesson Quiz 2 The table shows the top scores for girls in barrel racing at the 2004 National High School Rodeo finals. The data can be presented in a table

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared 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 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

4016 MATHEMATICS TOPIC 1: NUMBERS AND ALGEBRA SUB-TOPIC 1.11 MATRICES

4016 MATHEMATICS TOPIC 1: NUMBERS AND ALGEBRA SUB-TOPIC 1.11 MATRICES MATHEMATICS TOPIC : NUMBERS AND ALGEBRA SUB-TOPIC. MATRICES CONTENT OUTLINE. Display of information in the form of a matrix of any order. Interpreting the data in a given matrix. Product of a scalar quantity

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

1 Matrices and matrix algebra

1 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 information

Definition of Equality of Matrices. Example 1: Equality of Matrices. Consider the four matrices

Definition of Equality of Matrices. Example 1: Equality of Matrices. Consider the four matrices IT 131: Mathematics for Science Lecture Notes 3 Source: Larson, Edwards, Falvo (2009): Elementary Linear Algebra, Sixth Edition. Matrices 2.1 Operations with Matrices This section and the next introduce

More information

MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra

MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;

More information

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )

10. 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 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side

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

Matrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation.

Matrices. 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 information

Lecture 3: Matrix and Matrix Operations

Lecture 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 Scalar-Matrix operation: Scaling a matrix

More information

Linear Equations in Linear Algebra

Linear 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 information

v = ( 2)

v = ( 2) Chapter : Introduction to Vectors.. Vectors and linear combinations Let s begin by saying what vectors are: They are lists of numbers. If there are numbers in the list, there is a natural correspondence

More information

8 Matrices and operations on matrices

8 Matrices and operations on matrices AAC - Business Mathematics I Lecture #8, December 1, 007 Katarína Kálovcová 8 Matrices and operations on matrices Matrices: In mathematics, a matrix (plural matrices is a rectangular table of elements

More information

Systems of Linear Equations and Matrices

Systems 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 information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

Mathematics 13: Lecture 10

Mathematics 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 information

Linear Algebra (Review) Volker Tresp 2017

Linear Algebra (Review) Volker Tresp 2017 Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.

More information

UNIT 3 INTERSECTIONS OF LINES AND PLANES

UNIT 3 INTERSECTIONS OF LINES AND PLANES UNIT 3 INTERSECTIONS OF LINES AND PLANES UNIT 3 INTERSECTIONS OF LINES AND PLANES...1 VECTOR EQUATIONS OF LINES IN SCALAR EQUATION OF LINES IN EQUATIONS OF LINES IN 2...2 2...4 3...6 VECTOR AND SCALAR

More information

Systems of Linear Equations and Matrices

Systems 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 information

Elementary Row Operations on Matrices

Elementary 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 information

Image Registration Lecture 2: Vectors and Matrices

Image Registration Lecture 2: Vectors and Matrices Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this

More information

A summary of matrices and matrix math

A summary of matrices and matrix math A summary of matrices and matrix math Vince Cronin, Baylor University, reviewed and revised by Nancy West, Beth Pratt-Sitaula, and Shelley Olds. Total horizontal velocity vectors from three GPS stations

More information

MATRICES The numbers or letters in any given matrix are called its entries or elements

MATRICES The numbers or letters in any given matrix are called its entries or elements MATRICES A matrix is defined as a rectangular array of numbers. Examples are: 1 2 4 a b 1 4 5 A : B : C 0 1 3 c b 1 6 2 2 5 8 The numbers or letters in any given matrix are called its entries or elements

More information

Linear Algebra (Review) Volker Tresp 2018

Linear Algebra (Review) Volker Tresp 2018 Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c

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

Matrices. 1 a a2 1 b b 2 1 c c π

Matrices. 1 a a2 1 b b 2 1 c c π Matrices 2-3-207 A matrix is a rectangular array of numbers: 2 π 4 37 42 0 3 a a2 b b 2 c c 2 Actually, the entries can be more general than numbers, but you can think of the entries as numbers to start

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

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in

More information

n 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 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 information

Introduction to Determinants

Introduction to Determinants Introduction to Determinants For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix The matrix A is invertible if and only if.

More information

MATH 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 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 information

Introduction. Vectors and Matrices. Vectors [1] Vectors [2]

Introduction. 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 information

For comments, corrections, etc Please contact Ahnaf Abbas: Sharjah Institute of Technology. Matrices Handout #8.

For comments, corrections, etc Please contact Ahnaf Abbas: Sharjah Institute of Technology. Matrices Handout #8. Matrices Handout #8 Topic Matrix Definition A matrix is an array of numbers: a a2... a n a2 a22... a 2n A =.... am am2... amn Matrices are denoted by capital letters : A,B,C,.. Matrix size or rank is determined

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

Exercise Set Suppose that A, B, C, D, and E are matrices with the following sizes: A B C D E

Exercise Set Suppose that A, B, C, D, and E are matrices with the following sizes: A B C D E Determine the size of a given matrix. Identify the row vectors and column vectors of a given matrix. Perform the arithmetic operations of matrix addition, subtraction, scalar multiplication, and multiplication.

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

Multiplying matrices by diagonal matrices is faster than usual matrix multiplication.

Multiplying matrices by diagonal matrices is faster than usual matrix multiplication. 7-6 Multiplying matrices by diagonal matrices is faster than usual matrix multiplication. The following equations generalize to matrices of any size. Multiplying a matrix from the left by a diagonal matrix

More information

Section 12.4 Algebra of Matrices

Section 12.4 Algebra of Matrices 244 Section 2.4 Algebra of Matrices Before we can discuss Matrix Algebra, we need to have a clear idea what it means to say that two matrices are equal. Let's start a definition. Equal Matrices Two matrices

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

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

Lecture 3 Linear Algebra Background

Lecture 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 information

1. Vectors.

1. Vectors. 1. Vectors 1.1 Vectors and Matrices Linear algebra is concerned with two basic kinds of quantities: vectors and matrices. 1.1 Vectors and Matrices Scalars and Vectors - Scalar: a numerical value denoted

More information

CSCI 239 Discrete Structures of Computer Science Lab 6 Vectors and Matrices

CSCI 239 Discrete Structures of Computer Science Lab 6 Vectors and Matrices CSCI 239 Discrete Structures of Computer Science Lab 6 Vectors and Matrices This lab consists of exercises on real-valued vectors and matrices. Most of the exercises will required pencil and paper. Put

More information

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :

MAC 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

MATRICES. a m,1 a m,n A =

MATRICES. 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 information

A 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

A 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 information

Elementary maths for GMT

Elementary 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 information

Matrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix

Matrix 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 information

A 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

A 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 information

Review of linear algebra

Review of linear algebra Review of linear algebra 1 Vectors and matrices We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of

More information

An Introduction To Linear Algebra. Kuttler

An Introduction To Linear Algebra. Kuttler An Introduction To Linear Algebra Kuttler April, 7 Contents Introduction 7 F n 9 Outcomes 9 Algebra in F n Systems Of Equations Outcomes Systems Of Equations, Geometric Interpretations Systems Of Equations,

More information

Instruction: Operations with Matrices ( ) ( ) log 8 log 25 = If the corresponding elements do not equal, then the matrices are not equal.

Instruction: Operations with Matrices ( ) ( ) log 8 log 25 = If the corresponding elements do not equal, then the matrices are not equal. 7 Instruction: Operations with Matrices Two matrices are said to be equal if they have the same size and their corresponding elements are equal. For example, 3 ( ) ( ) ( ) ( ) log 8 log log log3 8 If the

More information

When two letters name a vector, the first indicates the and the second indicates the of the vector.

When two letters name a vector, the first indicates the and the second indicates the of the vector. 8-8 Chapter 8 Applications of Trigonometry 8.3 Vectors, Operations, and the Dot Product Basic Terminology Algeraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle etween Vectors

More information

x n -2.5 Definition A list is a list of objects, where multiplicity is allowed, and order matters. For example, as lists

x n -2.5 Definition A list is a list of objects, where multiplicity is allowed, and order matters. For example, as lists Vectors, Linear Combinations, and Matrix-Vector Mulitiplication In this section, we introduce vectors, linear combinations, and matrix-vector multiplication The rest of the class will involve vectors,

More information

Raphael Mrode. Training in quantitative genetics and genomics 30 May 10 June 2016 ILRI, Nairobi. Partner Logo. Partner Logo

Raphael Mrode. Training in quantitative genetics and genomics 30 May 10 June 2016 ILRI, Nairobi. Partner Logo. Partner Logo Basic matrix algebra Raphael Mrode Training in quantitative genetics and genomics 3 May June 26 ILRI, Nairobi Partner Logo Partner Logo Matrix definition A matrix is a rectangular array of numbers set

More information

Evaluating Determinants by Row Reduction

Evaluating 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 information

MATH 320, WEEK 7: Matrices, Matrix Operations

MATH 320, WEEK 7: Matrices, Matrix Operations MATH 320, WEEK 7: Matrices, Matrix Operations 1 Matrices We have introduced ourselves to the notion of the grid-like coefficient matrix as a short-hand coefficient place-keeper for performing Gaussian

More information

Matrix operations Linear Algebra with Computer Science Application

Matrix 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 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

Matrices BUSINESS MATHEMATICS

Matrices 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 information

Chapter 2 - Basics Structures MATH 213. Chapter 2: Basic Structures. Dr. Eric Bancroft. Fall Dr. Eric Bancroft MATH 213 Fall / 60

Chapter 2 - Basics Structures MATH 213. Chapter 2: Basic Structures. Dr. Eric Bancroft. Fall Dr. Eric Bancroft MATH 213 Fall / 60 MATH 213 Chapter 2: Basic Structures Dr. Eric Bancroft Fall 2013 Dr. Eric Bancroft MATH 213 Fall 2013 1 / 60 Chapter 2 - Basics Structures 2.1 - Sets 2.2 - Set Operations 2.3 - Functions 2.4 - Sequences

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

Elementary Linear Algebra

Elementary Linear Algebra Elementary Linear Algebra Linear algebra is the study of; linear sets of equations and their transformation properties. Linear algebra allows the analysis of; rotations in space, least squares fitting,

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

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 2 - Basics Structures

Chapter 2 - Basics Structures Chapter 2 - Basics Structures 2.1 - Sets Definitions and Notation Definition 1 (Set). A set is an of. These are called the or of the set. We ll typically use uppercase letters to denote sets: S, A, B,...

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

Math Studio College Algebra

Math Studio College Algebra Math 100 - Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014 Systems of Equations Systems of Equations A system of equations consists of Systems of Equations A system of

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

Matrices. Background mathematics review

Matrices. Background mathematics review Matrices Background mathematics review David Miller Matrices Matrix notation Background mathematics review David Miller Matrix notation A matrix is, first of all, a rectangular array of numbers An M N

More information

Section 5.5: Matrices and Matrix Operations

Section 5.5: Matrices and Matrix Operations Section 5.5 Matrices and Matrix Operations 359 Section 5.5: Matrices and Matrix Operations Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.

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

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

A primer on matrices

A primer on matrices A primer on matrices Stephen Boyd August 4, 2007 These notes describe the notation of matrices, the mechanics of matrix manipulation, and how to use matrices to formulate and solve sets of simultaneous

More information

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of

More information

Matrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form.

Matrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form. Matrices A matrix is a method of writing a set of numbers using rows and columns. 1 2 3 4 3 2 1 5 7 2 5 4 2 0 5 10 12 8 4 9 25 30 1 1 Reading Information from a Matrix Cells in a matrix can be referenced

More information

Numerical Analysis Lecture Notes

Numerical 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 information

MATRICES AND MATRIX OPERATIONS

MATRICES AND MATRIX OPERATIONS SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)

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

7.3. Determinants. Introduction. Prerequisites. Learning Outcomes

7.3. Determinants. Introduction. Prerequisites. Learning Outcomes Determinants 7.3 Introduction Among other uses, determinants allow us to determine whether a system of linear equations has a unique solution or not. The evaluation of a determinant is a key skill in engineering

More information

Chapter 6. Additional Topics in Trigonometry. 6.6 Vectors. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 6. Additional Topics in Trigonometry. 6.6 Vectors. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter 6 Additional Topics in Trigonometry 6.6 Vectors Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Obectives: Use magnitude and direction to show vectors are equal. Visualize scalar multiplication,

More information

MATH2210 Notebook 2 Spring 2018

MATH2210 Notebook 2 Spring 2018 MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................

More information