DOCSLIB.ORG

John Vince Fifth Edition

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
John Vince Fifth Edition Undergraduate Topics in Computer Science John Vince Mathematics for Computer Graphics Fifth Edition Undergraduate Topics in Computer Science Series editor Ian Mackie Advisory Board Samson Abramsky, University of Oxford, Oxford, UK Karin Breitman, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil Chris Hankin, Imperial College London, London, UK Dexter C. Kozen, Cornell University, Ithaca, USA Andrew Pitts, University of Cambridge, Cambridge, UK Hanne Riis Nielson, Technical University of Denmark, Kongens Lyngby, Denmark Steven S. Skiena, Stony Brook University, Stony Brook, USA Iain Stewart, University of Durham, Durham, UK Undergraduate Topics in Computer Science (UTiCS) delivers high-quality instructional content for undergraduates studying in all areas of computing and information science. From core foundational and theoretical material to final-year topics and applications, UTiCS books take a fresh, concise, and modern approach and are ideal for self-study or for a one- or two-semester course. The texts are all authored by established experts in their fields, reviewed by an international advisory board, and contain numerous examples and problems. Many include fully worked solutions. More information about this series at http://www.springer.com/series/7592 John Vince Mathematics for Computer Graphics Fifth Edition 123 John Vince Breinton UK ISSN 1863-7310 ISSN 2197-1781 (electronic) Undergraduate Topics in Computer Science ISBN 978-1-4471-7334-2 ISBN 978-1-4471-7336-6 (eBook) DOI 10.1007/978-1-4471-7336-6 Library of Congress Control Number: 2017943836 © Springer-Verlag London Ltd. 2001, 2006, 2010, 2014, 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag London Ltd. The registered company address is: 236 Gray’s Inn Road, London WC1X 8HB, United Kingdom This book is dedicated to my best friend and wife Heidi. Preface This book originally appeared as part of Springer’s excellent ‘Essential’ series and was revised to include chapters on analytical geometry, barycentric coordinates, and worked examples. The third edition included a new chapter on geometric algebra, which I have written about in my books Geometric Algebra for Computer Graphics and Geometric Algebra: An Algebraic System for Computer Games and Animation. In the fourth edition, I reviewed the entire book and included chapters on differ- ential and integral calculus, which I have written about in Calculus for Computer Graphics. This fifth edition includes some revisions and new content published in my recent book Foundation Mathematics for Computer Science. I have also redrawn all the figures in colour using Apple’s Pages and Grapher, which improve the book’s visual appearance. Whilst writing this book, I have borne in mind what it was like for me when I was studying different areas of mathematics for the first time. In spite of reading and rereading an explanation several times, it could take days before ‘the penny dropped’ and a concept became apparent. Hopefully, the reader will find the fol- lowing explanations useful in developing their understanding of these specific areas of mathematics and enjoy the sound of various pennies dropping! I would like to thank Beverley Ford, General Manager, Springer UK, and Helen Desmond, Editor for Computer Science, for persuading me to give up holidays and hobbies in order to complete this fifth edition! Breinton, UK John Vince vii Contents 1 Introduction ............................................ 1 1.1 Mathematics for Computer Graphics .................... 1 1.2 Understanding Mathematics ........................... 1 1.3 What Makes Mathematics Difficult?..................... 2 1.4 Does Mathematics Exist Outside Our Brains? ............. 2 1.5 Symbols and Notation ............................... 3 2 Numbers ............................................... 5 2.1 Introduction ....................................... 5 2.2 Background ....................................... 5 2.3 Counting ......................................... 5 2.4 Sets of Numbers.................................... 6 2.5 Zero ............................................. 7 2.6 Negative Numbers .................................. 8 2.6.1 The Arithmetic of Positive and Negative Numbers .................................. 9 2.7 Observations and Axioms............................. 10 2.7.1 Commutative Law........................... 10 2.7.2 Associative Law ............................ 10 2.7.3 Distributive Law ............................ 11 2.8 The Base of a Number System......................... 11 2.8.1 Background................................ 11 2.8.2 Octal Numbers ............................. 12 2.8.3 Binary Numbers ............................ 13 2.8.4 Hexadecimal Numbers ....................... 14 2.8.5 Adding Binary Numbers ...................... 17 2.8.6 Subtracting Binary Numbers ................... 18 2.9 Types of Numbers .................................. 19 2.9.1 Natural Numbers............................ 19 2.9.2 Integers ................................... 19 2.9.3 Rational Numbers ........................... 20 ix x Contents 2.9.4 Irrational Numbers .......................... 20 2.9.5 Real Numbers .............................. 20 2.9.6 Algebraic and Transcendental Numbers .......... 20 2.9.7 Imaginary Numbers.......................... 21 2.9.8 Complex Numbers .......................... 25 2.9.9 Transcendental and Algebraic Numbers .......... 27 2.9.10 Infinity ................................... 27 2.10 Summary ......................................... 28 2.11 Worked Examples .................................. 29 2.11.1 Algebraic Expansion ......................... 29 2.11.2 Binary Subtraction .......................... 29 2.11.3 Complex Numbers .......................... 29 2.11.4 Complex Rotation ........................... 30 3 Algebra ................................................ 31 3.1 Introduction ....................................... 31 3.2 Background ....................................... 32 3.2.1 Solving the Roots of a Quadratic Equation ........ 33 3.3 Indices ........................................... 37 3.3.1 Laws of Indices............................. 37 3.4 Logarithms ........................................ 38 3.5 Further Notation.................................... 40 3.6 Functions ......................................... 40 3.6.1 Explicit and Implicit Equations ................. 40 3.6.2 Function Notation ........................... 41 3.6.3 Intervals .................................. 42 3.6.4 Function Domains and Ranges ................. 43 3.6.5 Odd and Even Functions...................... 44 3.6.6 Power Functions ............................ 45 3.7 Summary ......................................... 46 3.8 Worked Examples .................................. 46 3.8.1 Algebraic Manipulation....................... 46 3.8.2 Solving a Quadratic Equation .................. 47 3.8.3 Factorising ................................ 49 4 Trigonometry ........................................... 51 4.1 Introduction ....................................... 51 4.2 Background ....................................... 51 4.3 Units of Angular Measurement ........................ 51 4.4 The Trigonometric Ratios............................. 52 4.4.1 Domains and Ranges ........................ 55 4.5 Inverse Trigonometric Ratios .......................... 55 4.6 Trigonometric Identities .............................. 57 4.7 The Sine Rule ..................................... 58 4.8 The Cosine Rule ................................... 58 Contents xi 4.9 Compound-Angle Identities ........................... 59 4.9.1 Double-Angle Identities ...................... 60 4.9.2 Multiple-Angle Identities...................... 61 4.9.3 Half-Angle Identities......................... 62 4.10 Perimeter Relationships .............................. 62 4.11 Summary ......................................... 63 5 Coordinate Systems ...................................... 65 5.1 Introduction ....................................... 65 5.2 Background ....................................... 65 5.3 The Cartesian Plane ................................. 66 5.4 Function Graphs.................................... 66 5.5 Shape Representation ................................ 67 5.5.1
Load more

Recommended publications
  • Final Exam Outline 1.1, 1.2 Scalars and Vectors in R N. the Magnitude
    Final Exam Outline 1.1, 1.2 Scalars and vectors in Rn. The magnitude of a vector. Row and column vectors. Vector addition. Scalar multiplication. The zero vector. Vector subtraction. Distance in Rn. Unit vectors. Linear combinations. The dot product. The angle between two vectors. Orthogonality. The CBS and triangle inequalities. Theorem 1.5. 2.1, 2.2 Systems of linear equations. Consistent and inconsistent systems. The coefficient and augmented matrices. Elementary row operations. The echelon form of a matrix. Gaussian elimination. Equivalent matrices. The reduced row echelon form. Gauss-Jordan elimination. Leading and free variables. The rank of a matrix. The rank theorem. Homogeneous systems. Theorem 2.2. 2.3 The span of a set of vectors. Linear dependence. Linear dependence and independence. Determining n whether a set {~v1, . ,~vk} of vectors in R is linearly dependent or independent. 3.1, 3.2 Matrix operations. Scalar multiplication, matrix addition, matrix multiplication. The zero and identity matrices. Partitioned matrices. The outer product form. The standard unit vectors. The transpose of a matrix. A system of linear equations in matrix form. Algebraic properties of the transpose. 3.3 The inverse of a matrix. Invertible (nonsingular) matrices. The uniqueness of the solution to Ax = b when A is invertible. The inverse of a nonsingular 2 × 2 matrix. Elementary matrices and EROs. Equivalent conditions for invertibility. Using Gauss-Jordan elimination to invert a nonsingular matrix. Theorems 3.7, 3.9. n n n 3.5 Subspaces of R . {0} and R as subspaces of R . The subspace span(v1,..., vk). The the row, column and null spaces of a matrix.
    [Show full text]
  • Introduction to MATLAB®
    Introduction to MATLAB® Lecture 2: Vectors and Matrices Introduction to MATLAB® Lecture 2: Vectors and Matrices 1 / 25 Vectors and Matrices Vectors and Matrices Vectors and matrices are used to store values of the same type A vector can be either column vector or a row vector. Matrices can be visualized as a table of values with dimensions r × c (r is the number of rows and c is the number of columns). Introduction to MATLAB® Lecture 2: Vectors and Matrices 2 / 25 Vectors and Matrices Creating row vectors Place the values that you want in the vector in square brackets separated by either spaces or commas. e.g 1 >> row vec=[1 2 3 4 5] 2 row vec = 3 1 2 3 4 5 4 5 >> row vec=[1,2,3,4,5] 6 row vec= 7 1 2 3 4 5 Introduction to MATLAB® Lecture 2: Vectors and Matrices 3 / 25 Vectors and Matrices Creating row vectors - colon operator If the values of the vectors are regularly spaced, the colon operator can be used to iterate through these values. (first:last) produces a vector with all integer entries from first to last e.g. 1 2 >> row vec = 1:5 3 row vec = 4 1 2 3 4 5 Introduction to MATLAB® Lecture 2: Vectors and Matrices 4 / 25 Vectors and Matrices Creating row vectors - colon operator A step value can also be specified with another colon in the form (first:step:last) 1 2 >>odd vec = 1:2:9 3 odd vec = 4 1 3 5 7 9 Introduction to MATLAB® Lecture 2: Vectors and Matrices 5 / 25 Vectors and Matrices Exercise In using (first:step:last), what happens if adding the step value would go beyond the range specified by last? e.g: 1 >>v = 1:2:6 Introduction to MATLAB® Lecture 2: Vectors and Matrices 6 / 25 Vectors and Matrices Exercise Use (first:step:last) to generate the vector v1 = [9 7 5 3 1 ]? Introduction to MATLAB® Lecture 2: Vectors and Matrices 7 / 25 Vectors and Matrices Creating row vectors - linspace function linspace (Linearly spaced vector) 1 >>linspace(x,y,n) linspace creates a row vector with n values in the inclusive range from x to y.
    [Show full text]
  • Math 102 -- Linear Algebra I -- Study Guide
    Math 102 Linear Algebra I Stefan Martynkiw These notes are adapted from lecture notes taught by Dr.Alan Thompson and from “Elementary Linear Algebra: 10th Edition” :Howard Anton. Picture above sourced from (http://i.imgur.com/RgmnA.gif) 1/52 Table of Contents Chapter 3 – Euclidean Vector Spaces.........................................................................................................7 3.1 – Vectors in 2-space, 3-space, and n-space......................................................................................7 Theorem 3.1.1 – Algebraic Vector Operations without components...........................................7 Theorem 3.1.2 .............................................................................................................................7 3.2 – Norm, Dot Product, and Distance................................................................................................7 Definition 1 – Norm of a Vector..................................................................................................7 Definition 2 – Distance in Rn......................................................................................................7 Dot Product.......................................................................................................................................8 Definition 3 – Dot Product...........................................................................................................8 Definition 4 – Dot Product, Component by component..............................................................8
    [Show full text]
  • Lesson 1 Introduction to MATLAB
    Math 323 Linear Algebra and Matrix Theory I Fall 1999 Dr. Constant J. Goutziers Department of Mathematical Sciences [email protected] Lesson 1 Introduction to MATLAB MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. This allows easy solution of many technical computing problems, especially those with matrix and vector formulations. For many colleges and universities MATLAB is the Linear Algebra software of choice. The name MATLAB stands for matrix laboratory. 1.1 Row and Column vectors, Formatting In linear algebra we make a clear distinction between row and column vectors. • Example 1.1.1 Entering a row vector. The components of a row vector are entered inside square brackets, separated by blanks. v=[1 2 3] v = 1 2 3 • Example 1.1.2 Entering a column vector. The components of a column vector are also entered inside square brackets, but they are separated by semicolons. w=[4; 5/2; -6] w = 4.0000 2.5000 -6.0000 • Example 1.1.3 Switching between a row and a column vector, the transpose. In MATLAB it is possible to switch between row and column vectors using an apostrophe. The process is called "taking the transpose". To illustrate the process we print v and w and generate the transpose of each of these vectors. Observe that MATLAB commands can be entered on the same line, as long as they are separated by commas. v, vt=v', w, wt=w' v = 1 2 3 vt = 1 2 3 w = 4.0000 2.5000 -6.0000 wt = 4.0000 2.5000 -6.0000 • Example 1.1.4 Formatting.
    [Show full text]
  • Linear Algebra I: Vector Spaces A
    Linear Algebra I: Vector Spaces A 1 Vector spaces and subspaces 1.1 Let F be a field (in this book, it will always be either the field of reals R or the field of complex numbers C). A vector space V D .V; C; o;˛./.˛2 F// over F is a set V with a binary operation C, a constant o and a collection of unary operations (i.e. maps) ˛ W V ! V labelled by the elements of F, satisfying (V1) .x C y/ C z D x C .y C z/, (V2) x C y D y C x, (V3) 0 x D o, (V4) ˛ .ˇ x/ D .˛ˇ/ x, (V5) 1 x D x, (V6) .˛ C ˇ/ x D ˛ x C ˇ x,and (V7) ˛ .x C y/ D ˛ x C ˛ y. Here, we write ˛ x and we will write also ˛x for the result ˛.x/ of the unary operation ˛ in x. Often, one uses the expression “multiplication of x by ˛”; but it is useful to keep in mind that what we really have is a collection of unary operations (see also 5.1 below). The elements of a vector space are often referred to as vectors. In contrast, the elements of the field F are then often referred to as scalars. In view of this, it is useful to reflect for a moment on the true meaning of the axioms (equalities) above. For instance, (V4), often referred to as the “associative law” in fact states that the composition of the functions V ! V labelled by ˇ; ˛ is labelled by the product ˛ˇ in F, the “distributive law” (V6) states that the (pointwise) sum of the mappings labelled by ˛ and ˇ is labelled by the sum ˛ C ˇ in F, and (V7) states that each of the maps ˛ preserves the sum C.
    [Show full text]
  • Week 1 – Vectors and Matrices
    Week 1 – Vectors and Matrices Richard Earl∗ Mathematical Institute, Oxford, OX1 2LB, October 2003 Abstract Algebra and geometry of vectors. The algebra of matrices. 2x2 matrices. Inverses. Determinants. Simultaneous linear equations. Standard transformations of the plane. Notation 1 The symbol R2 denotes the set of ordered pairs (x, y) – that is the xy-plane. Similarly R3 denotes the set of ordered triples (x, y, z) – that is, three-dimensional space described by three co-ordinates x, y and z –andRn denotes a similar n-dimensional space. 1Vectors A vector can be thought of in two different ways. Let’s for the moment concentrate on vectors in the xy-plane. From one point of view a vector is just an ordered pair of numbers (x, y). We can associate this • vector with the point in R2 which has co-ordinates x and y. We call this vector the position vector of the point. From the second point of view a vector is a ‘movement’ or translation. For example, to get from • the point (3, 4) to the point (4, 5) we need to move ‘one to the right and one up’; this is the same movement as is required to move from ( 2, 3) to ( 1, 2) or from (1, 2) to (2, 1) . Thinking about vectors from this second point of− view,− all three− of− these movements− are the− same vector, because the same translation ‘one right, one up’ achieves each of them, even though the ‘start’ and ‘finish’ are different in each case. We would write this vector as (1, 1) .
    [Show full text]
  • Linear Algebra: Matrices, Vectors, Determinants. Linear Systems
    c07.qxd 10/28/10 7:30 PM Page 256 CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra is a fairly extensive subject that covers vectors and matrices, determinants, systems of linear equations, vector spaces and linear transformations, eigenvalue problems, and other topics. As an area of study it has a broad appeal in that it has many applications in engineering, physics, geometry, computer science, economics, and other areas. It also contributes to a deeper understanding of mathematics itself. Matrices, which are rectangular arrays of numbers or functions, and vectors are the main tools of linear algebra. Matrices are important because they let us express large amounts of data and functions in an organized and concise form. Furthermore, since matrices are single objects, we denote them by single letters and calculate with them directly. All these features have made matrices and vectors very popular for expressing scientific and mathematical ideas. The chapter keeps a good mix between applications (electric networks, Markov processes, traffic flow, etc.) and theory. Chapter 7 is structured as follows: Sections 7.1 and 7.2 provide an intuitive introduction to matrices and vectors and their operations, including matrix multiplication. The next block of sections, that is, Secs. 7.3–7.5 provide the most important method for solving systems of linear equations by the Gauss elimination method. This method is a cornerstone of linear algebra, and the method itself and variants of it appear in different areas of mathematics and in many applications. It leads to a consideration of the behavior of solutions and concepts such as rank of a matrix, linear independence, and bases.
    [Show full text]
  • Chapter 9 Matrices and Determinants
    Chapter 9 222 Matrices and Determinants Chapter 9 Matrices and Determinants 9.1 Introduction: In many economic analysis, variables are assumed to be related by sets of linear equations. Matrix algebra provides a clear and concise notation for the formulation and solution of such problems, many of which would be complicated in conventional algebraic notation. The concept of determinant and is based on that of matrix. Hence we shall first explain a matrix. 9.2 Matrix: A set of mn numbers (real or complex), arranged in a rectangular formation (array or table) having m rows and n columns and enclosed by a square bracket [ ] is called mn matrix (read “m by n matrix”) . An mn matrix is expressed as a11 a 12 a 1n a21 a 22 a 2n A= am1 a m2 a mn The letters aij stand for real numbers. Note that aij is the element in the ith row and jth column of the matrix .Thus the matrix A is sometimes denoted by simplified form as (aij) or by {aij} i.e., A = (aij) Matrices are usually denoted by capital letters A, B, C etc and its elements by small letters a, b, c etc. Order of a Matrix: The order or dimension of a matrix is the ordered pair having as first component the number of rows and as second component the number of columns in the matrix. If there are 3 rows and 2 columns in a matrix, then its order is written as (3, 2) or (3 x 2) read as three by two.
    [Show full text]
  • Quantum Algorithms Via Linear Algebra: a Primer / Richard J
    QUANTUM ALGORITHMS VIA LINEAR ALGEBRA A Primer Richard J. Lipton Kenneth W. Regan The MIT Press Cambridge, Massachusetts London, England c 2014 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or sales promotional use. For information, please email special [email protected]. This book was set in Times Roman and Mathtime Pro 2 by the authors, and was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Lipton, Richard J., 1946– Quantum algorithms via linear algebra: a primer / Richard J. Lipton and Kenneth W. Regan. p. cm. Includes bibliographical references and index. ISBN 978-0-262-02839-4 (hardcover : alk. paper) 1. Quantum computers. 2. Computer algorithms. 3. Algebra, Linear. I. Regan, Kenneth W., 1959– II. Title QA76.889.L57 2014 005.1–dc23 2014016946 10987654321 We dedicate this book to all those who helped create and nourish the beautiful area of quantum algorithms, and to our families who helped create and nourish us. RJL and KWR Contents Preface xi Acknowledgements xiii 1 Introduction 1 1.1 The Model 2 1.2 The Space and the States 3 1.3 The Operations 5 1.4 Where Is the Input? 6 1.5 What Exactly Is the Output? 7 1.6 Summary and Notes 8 2 Numbers and Strings 9 2.1 Asymptotic Notation
    [Show full text]
  • Computational Foundations of Cognitive Science Lecture 11: Matrices in Matlab
    Basic Matrix Operations Special Matrices Matrix Products Transpose, Inner and Outer Product Computational Foundations of Cognitive Science Lecture 11: Matrices in Matlab Frank Keller School of Informatics University of Edinburgh [email protected] February 23, 2010 Frank Keller Computational Foundations of Cognitive Science 1 Basic Matrix Operations Special Matrices Matrix Products Transpose, Inner and Outer Product 1 Basic Matrix Operations Sum and Difference Size; Product with Scalar 2 Special Matrices Zero and Identity Matrix Diagonal and Triangular Matrices Block Matrices 3 Matrix Products Row and Column Vectors Mid-lecture Problem Matrix Product Product with Vector 4 Transpose, Inner and Outer Product Transpose Symmetric Matrices Inner and Outer Product Reading: McMahon, Ch. 2 Frank Keller Computational Foundations of Cognitive Science 2 Basic Matrix Operations Special Matrices Sum and Difference Matrix Products Size; Product with Scalar Transpose, Inner and Outer Product Sum and Difference In Matlab, matrices are input as lists of numbers; columns are separated by spaces or commas, rows by semicolons or newlines: > A = [2, 1, 0, 3; -1, 0, 2, 4; 4, -2, 7, 0]; > B = [-4, 3, 5, 1 2, 2, 0, -1 3, 2, -4, 5]; > C = [1 1; 2 2]; The sum and difference of two matrices can be computed using the operators + and -: > disp(A + B); -2 4 5 4 1 2 2 3 7 0 3 5 Frank Keller Computational Foundations of Cognitive Science 3 Basic Matrix Operations Special Matrices Sum and Difference Matrix Products Size; Product with Scalar Transpose, Inner and Outer Product
    [Show full text]
  • INTRODUCTION to MATRIX ALGEBRA 1.1. Definition of a Matrix
    INTRODUCTION TO MATRIX ALGEBRA 1. DEFINITION OF A MATRIX AND A VECTOR 1.1. Definition of a matrix. A matrix is a rectangular array of numbers arranged into rows and columns. It is written as a11 a12 ... a1n a a ... a 21 22 2n .. (1) .. .. am1 am2 ... amn The above array is called an m by n (m x n) matrix since it has m rows and n columns. Typically upper-case letters are used to denote a matrix and lower case letters with subscripts the elements. The matrix A is also often denoted A = kaijk (2) 1.2. Definition of a vector. A vector is a n-tuple of numbers. In R2 a vector would be an ordered 3 n pair of numbers {x, y}.InR a vector is a 3-tuple, i.e., {x1,x2,x3}. Similarly for R . Vectors are usually denoted by lower case letters such as a or b, or more formally ~a or ~b. 1.3. Row and column vectors. 1.3.1. Row vector. A matrix with one row and n columns (1xn) is called a row vector. It is usually written ~x 0 or 0 ~x = x1,x2,x3, ..., xn (3) The use of the prime 0 symbol indicates we are writing the n-tuple horizontally as if it were the row of a matrix. Note that each row of a matrix is a row vector. 1.3.2. Column vector. A matrix with one column and n rows (nx1) is called a column vector. It is written as x1 x 2 x3 ~x = .
    [Show full text]
  • 1.1. Elementary Matrices, Row and Column Transformations. We Have
    1. ROW OPERATION AND COLUMN OPERATIONS 1.1. Elementary Matrices, Row and Column Transformations. We have already studied in the last section that we can view the matrix multiplication as linear combi- nation of column vectors of the first matrix, or row vectors of the second matrix. And the coefficient of matrix multiplication is exactly given by other matrix. This shows that to understand matrix multiplication, we have to study the linear combination of row and column vectors. In this section, we will study the most basic linear combination of rows and columns, row and column transformation. 1.1.1. Elementary Row transformation. We have three types of row transformation. row switching This transformation swiches two row of matrix. 0 1 4 8 1 0 3 3 5 1 r1$r3 Switch the 1st and 3rd row of matrix @ 2 0 9 A −−−−! @ 2 0 9 A 3 3 5 1 4 8 row multiplying This transformation multiplies some row with a non-zero scalar λ 0 1 4 8 1 0 1 4 8 1 2×r2 Multiply the 2nd row of matrix by 2 : @ 2 0 9 A −−−! @ 4 0 18 A 3 3 5 3 3 5 row adding In this transformation, we multiply some row by a scalar, but add that into another row. 0 1 4 8 1 0 1 4 8 1 r3+2×r2 Add twice of the 2nd row to the 3rd row : @ 2 0 9 A −−−−−! @ 2 0 9 A 3 3 5 7 3 23 Caution: Write 2 × r instead of r × 2, the reason for that is simple, because scalar is 1 × 1 matrix.
    [Show full text]