ICS 6N Computational Linear Algebra Matrix Algebra
|
|
- Mervyn Owen
- 6 years ago
- Views:
Transcription
1 ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, / 24
2 Matrix Consider an m n matrix a 11 a a 1n a 21 a a 1n A = = [ ] a 1 a 2... a n a m1 a m2... a mn a ij is the scalar entry in the ith row and jth column, called the (i, j)-entry. Each column is a vector in R m. Two matrices are equal if they have the same size and the corresponding entries are equal a 11, a 22,... are called the diagonal entries A is called diagonal if all non-diagonal entries are zero The identity matrix I n is a square diagonal matrix with diagonal being 1 The zero matrix is a matrix in which all entries are zero, written as 0. Xiaohui Xie (UCI) ICS 6N February 2, / 24
3 Matrix operations Given two m n matrices A and B, Sum: A + B is an m n matrix whose (i, j)-entry is a ij + b ij Multiplication by a scalar: ra = Ar is an m n matrix whose (i, j)-entry is ra ij, where r is a scalar. Matrix vector product: Ax = x 1 a 1 + x 2 a x n a n Xiaohui Xie (UCI) ICS 6N February 2, / 24
4 Examples Given A = A + B 2A A + C [ ] 1 2 3, B = [ ] 1 0 1, C = [ ] 1 0, compute 0 1 Xiaohui Xie (UCI) ICS 6N February 2, / 24
5 Examples [ ] Given A =, B = [ ] A + B = [ ] A = A + C = Not defined. [ ] 1 0 1, C = [ ] 1 0, 0 1 Xiaohui Xie (UCI) ICS 6N February 2, / 24
6 Properties of matrix operations Given A, B, C matrices of the same size, and scalars r and s, (A + B) + C = A + (B + C) A + B = B + A A + 0 = A r(a + B) = ra + rb (r + s)a = ra + sa r(sa) = (rs)a Xiaohui Xie (UCI) ICS 6N February 2, / 24
7 Matrix Multiplication When a matrix B multiplies a vector x, it transforms x into the vector Bx. If this vector is then multiplied in turn by a matrix A, the resulting vector is A(Bx). Thus A(Bx) is produced from x by a composition of mappings the linear transformations. Our goal is to represent this composite mapping as multiplication by a single matrix, denoted by AB, so that A(Bx) = (AB)x. Xiaohui Xie (UCI) ICS 6N February 2, / 24
8 Matrix Multiplication Suppose A is m n, B is n p, and x is in R p Denote B = [ b 1 b 2... b p ]. Xiaohui Xie (UCI) ICS 6N February 2, / 24
9 Matrix Multiplication Suppose A is m n, B is n p, and x is in R p Bx is a vector in R n, A(Bx) is a vector in R m Denote B = [ b 1 b 2... b p ]. Then Bx = x1 b 1 + x 2 b x p b p A(Bx) = A(x 1 b 1 + x 2 b x p b p ) = A(x 1 b 1 ) + A(x 2 b 2 ) A(x p b p ) = x 1 (Ab 1 ) + x 2 (Ab 2 ) x p (Ab p ) linear combination = [ Ab 1 Ab 2 Ab p ] x So AB = [ Ab 1 Ab 2 Ab p ], an m p matrix. Xiaohui Xie (UCI) ICS 6N February 2, / 24
10 MATRIX MULTIPLICATION Definition: If A is an m n matrix, and if B is an n p matrix with columns b 1,, b p, then the product AB is the m p matrix whose columns are Ab 1,, Ab p. That is AB = [ Ab 1 Ab 2 Ab p ] Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B. Xiaohui Xie (UCI) ICS 6N February 2, / 24
11 Example Given A = [ ] and B = 1 1, compute AB Xiaohui Xie (UCI) ICS 6N February 2, / 24
12 Example [ ] Given A = and B = 1 1, compute AB Solution: AB = Ab 1 Ab 2 [ ] 1 [ ] [ ] Ab 1 = =, Ab = 1 1 = [ ] 1 3 So AB = 1 3 [ ] 3 3 Xiaohui Xie (UCI) ICS 6N February 2, / 24
13 Row-column rule for computing AB Now let s check the (i, j)-entry of AB: (AB) ij = the i-th entry of the j-th column = the i-th entry of Ab j = b j (the i-th row of A) = a i1 b 1j + a i2 b 2j a in b nj The (i, j)-entry of AB is the sum of the products of corresponding entries from row i of A and column j of B (AB) ij = a i1 b 1j + a i2 b 2j a in b nj = n a ik b kj k=1 Xiaohui Xie (UCI) ICS 6N February 2, / 24
14 Example Given A = [ ] , B = 1 0, compute AB Xiaohui Xie (UCI) ICS 6N February 2, / 24
15 Example Given A = Solution: [ ] , B = 1 0, compute AB Are the sizes consistent? Yes Based on definition, AB = Ab 1 Ab 2 = Or we can also calculate this entry by entry (AB) 11 = [ ] [ ] = 3 (AB) 21 = [ ] [ ] = 9 And so[ on until ] we get 3 2 AB = 9 2 [ ] Xiaohui Xie (UCI) ICS 6N February 2, / 24
16 Special Cases An nx1 matrix can be viewed as a vector in R n (column vector) A row vector can be viewed as a 1xn matrix. (Dot product) A row vector times a column vector produces a scalar if they are of the same size. b 1 [ ] b 2 a1 a 2... a n. = a 1b 1 + a 2 b a n b n b n Xiaohui Xie (UCI) ICS 6N February 2, / 24
17 Special Cases (Out product) A column vector times a row vector produces a matrix. a 1 a 1 b 1 a 1 b 2... a 1 b n a 2. [ ] a 2 b 1 a 2 b 2... a 2 b n b 1 b 2... b n = a m a m b 1 a m b 2... a m b n Xiaohui Xie (UCI) ICS 6N February 2, / 24
18 Special cases Let A be an mxn matrix, AI n = A = I m A A0 = 0. Xiaohui Xie (UCI) ICS 6N February 2, / 24
19 Theorems If the sizes are consistent a) (AB)C = A(BC) b) A(B + C) = AB + AC c) (B + C)A = BA + BC d) (ra)b = A(rB) e) I m A = AI n Xiaohui Xie (UCI) ICS 6N February 2, / 24
20 Warnings AB BA in general. They are not even of the same size! Example Even[ if they ] are the [ same ] size it is in general not true A =, B = [ ] [ ] AB =, BA = If AB = BA then A and B are commutable, but in general they are not. Xiaohui Xie (UCI) ICS 6N February 2, / 24
21 Warnings If AB = AC and A 0, we cannot conclude B = C Example Even[ if they ] are the [ same ] size it is[ in general ] not true A =, B =, C = [ ] [ ] AB =, BA = But we can clearly see B C Xiaohui Xie (UCI) ICS 6N February 2, / 24
22 Powers of a matrix Definition: If A is an n n matrix and if k is a positive integer, then A k denotes the product of k copies of A. A k = A A (k times) A 0 = I by convention. Xiaohui Xie (UCI) ICS 6N February 2, / 24
23 Transpose Given an mxn matrix A, the transpose of A is the nxm matrix, denoted by A T, whose columns are formed from the corresponding rows of A. a 11 a a 1n a 11 a a n1 a 21 a a 2n If A =......, then a 12 a a n2 AT = a m1 a m2... a mn a 1m a 2m... a nm (A T ) ij = a ji Xiaohui Xie (UCI) ICS 6N February 2, / 24
24 Properties of matrix transpose If the sizes are consistent (A T ) T = A (A + B) T = A T + B T (ra) T = ra T (ra)b = A(rB) (AB) T = B T A T (note the reverse order!) Xiaohui Xie (UCI) ICS 6N February 2, / 24
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 informationMatrix 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 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 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 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 informationMatrix 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.
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
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 Row-Column 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 informationICS 6N Computational Linear Algebra Vector Equations
ICS 6N Computational Linear Algebra Vector Equations Xiaohui Xie University of California, Irvine xhx@uci.edu January 17, 2017 Xiaohui Xie (UCI) ICS 6N January 17, 2017 1 / 18 Vectors in R 2 An example
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 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 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 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 top-selling items are peanuts, hot dogs and soda.
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 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side
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 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 i-th row and j-th column
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 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 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 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 Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)
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 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 informationAnnouncements Monday, October 02
Announcements Monday, October 02 Please fill out the mid-semester 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 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 informationCS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:
CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 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 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 Scalar-Matrix operation: Scaling a matrix
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 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 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 informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
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 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 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 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 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 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 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 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 informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationICS 6N Computational Linear Algebra Vector Space
ICS 6N Computational Linear Algebra Vector Space Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 24 Vector Space Definition: A vector space is a non empty set V of
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 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 informationMATRICES 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 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 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 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 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 informationBasic Concepts in Linear Algebra
Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear
More informationMatrices 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 informationLinear Equations and Matrix
1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
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 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 informationMatrix & 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 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 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 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 informationFoundations of Cryptography
Foundations of Cryptography Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi Department of Mathematics and Statistics University of Turku 2017 Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi
More informationWhat is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010
What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 Scott Strong sstrong@mines.edu Colorado School of Mines What is the Matrix? p. 1/20 Overview/Keywords/References Advanced
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 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 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 informationLinear algebra comments. Sophie Marques
Linear algebra comments Sophie Marques Friday 9 th October, 2015 2 Of course this does not cover all the class notes and it is not enough to do the midterm. It is just a way to extract the very very important
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 informationICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization
ICS 6N Computational Linear Algebra Symmetric Matrices and Orthogonal Diagonalization Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 21 Symmetric matrices An n n
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 informationMath 360 Linear Algebra Fall Class Notes. a a a a a a. a a a
Math 360 Linear Algebra Fall 2008 9-10-08 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 informationGraduate 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 informationUndergraduate 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 information10: Representation of point group part-1 matrix algebra CHEMISTRY. PAPER No.13 Applications of group Theory
1 Subject Chemistry Paper No and Title Module No and Title Module Tag Paper No 13: Applications of Group Theory CHE_P13_M10 2 TABLE OF CONTENTS 1. Learning outcomes 2. Introduction 3. Definition of a matrix
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 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 2-dimensional and 3-dimensional graphics programming
More information3 a 21 a a 2N. 3 a 21 a a 2M
APPENDIX: MATHEMATICS REVIEW G 12.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers 2 A = 6 4 a 11 a 12... a 1N a 21 a 22... a 2N. 7..... 5 a M1 a M2...
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 informationMATH2210 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 informationChapter 2: Matrix Algebra
Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry
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 informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationBasic Linear Algebra in MATLAB
Basic Linear Algebra in MATLAB 9.29 Optional Lecture 2 In the last optional lecture we learned the the basic type in MATLAB is a matrix of double precision floating point numbers. You learned a number
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 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 informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationPOLI270 - 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 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 informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationReview of Basic Concepts in Linear Algebra
Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra
More informationTopics. 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 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 informationMTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018)
MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) COURSEWORK 3 SOLUTIONS Exercise ( ) 1. (a) Write A = (a ij ) n n and B = (b ij ) n n. Since A and B are diagonal, we have a ij = 0 and
More informationMath 1553 Introduction to Linear Algebra
Math 1553 Introduction to Linear Algebra Lecture Notes Chapter 2 Matrix Algebra School of Mathematics The Georgia Institute of Technology Math 1553 Lecture Notes for Chapter 2 Introduction, Slide 1 Section
More informationMATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.
MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University February 6, 2018 Linear Algebra (MTH
More informationMatrices 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 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 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 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 informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 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 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 informationMatrix Algebra. Learning Objectives. Size of Matrix
Matrix Algebra 1 Learning Objectives 1. Find the sum and difference of two matrices 2. Find scalar multiples of a matrix 3. Find the product of two matrices 4. Find the inverse of a matrix 5. Solve a system
More informationSection 1.6. M N = [a ij b ij ], (1.6.2)
The Calculus of Functions of Several Variables Section 16 Operations with Matrices In the previous section we saw the important connection between linear functions and matrices In this section we will
More informationChapter 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 informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces Per-Olof 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 information