Introduction to Linear Functions

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An Introduction to Linear Algebra Barry M. Wise and Neal B. Gallagher Eigenvector Research, Inc. 830 Wapato Lake Road Manson, WA 98831 USA bmw@eigenvector.com Linear algebra is the language of chemometrics. One cannot expect to truly understand most chemometric techniques without a basic understanding of linear algebra. This article reviews the basics of linear algebra and provides the reader with the foundation required for understanding most chemometrics literature. It is presented in a rather dense fashion: no proofs are given and there is little discussion of the theoretical implications of the theorems and results presented. The goal has been to condense into as few pages as possible the aspects of linear algebra used in most chemometric methods. Readers who are somewhat familiar with linear algebra may find this article to be a good quick review. Those totally unfamiliar with linear algebra should consider spending some time with a linear algebra text. In particular, those by Gilbert Strang are particularly easy to read and understand. Several of the numerical examples in this section are adapted from Strang’s Linear Algebra and Its Applications, Second Edition (Academic Press, 1980). MATLAB (The MathWorks, Inc., Natick MA) commands for performing the operations listed are also included; the reader is encouraged to run the examples presented in the text. Those unfamiliar with MATLAB may wish to read the first few sections of the tutorial chapter of the MATLAB User’s Guide. Scalars, Vectors and Matrices A scalar is a mathematical quantity that is completely described by a magnitude, i.e. a single number. Scalar variables are generally denoted by lowercase letters, e.g. a. Examples of scalar variables include temperature, density, pressure and flow. In MATLAB, a value can be assigned to a scalar at the command line, e.g. »a = 5; Here we have used the semicolon operator to suppress the echo of the result. Without this semicolon MATLAB would display the result of the assignment: »a = 5 a = 5 A vector is a mathematical quantity that is completely described by its magnitude and direction. An example of a three dimensional column vector might be b =       4 3 5 (1) Vectors are generally denoted by bold lowercase letters. (In MATLAB no distinction is made between the notation for scalars, vectors and matrices: they can be upper or lower case and bold letters are not used.) In MATLAB, this vector could be entered at the command in one of several ways. One way would be to enter the vector with an element on each line, like this: »b = [4 3 5] b = 4 3 5 Another way to enter the vector would be to use the semicolon to tell MATLAB that each line was completed. For instance »b = [4; 3; 5]; produces the same result as above. This vector is represented geometrically in Figure 1, where the three components 4, 3 and 5 are the coordinates of a point in three-dimensional space. Any vector b can be represented by a point in space; there is a perfect match between points and vectors. One can choose to think of a vector as the arrow, the point in space, or as the three numbers which describe the point. In a problem with 400 dimensions (such as in spectroscopy), it is probably easiest to consider the 400 numbers. The transpose of a column vector is a row vector and vice-versa. The transpose is generally indicated by a superscript T, i.e. T , though in some instances, including MATLAB, an apostrophe (') will be used. For example b T = [] 4 3 5 (2) In MATLAB, we could easily assign b T to Introduction to Linear Functions Introduction to Linear Functions By: OpenStaxCollege 1/3 Introduction to Linear Functions A bamboo forest in China (credit: “JFXie”/Flickr) Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later Although it may seem incredible, this can happen with certain types of bamboo species These members of the grass family are the fastest-growing plants in the world One species of bamboo has been observed to grow nearly 1.5 inches every hour http://www.guinnessworldrecords.com/records-3000/fastest-growing-plant/ 2/3 Introduction to Linear Functions In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function Recall from Functions and Function Notation that a function is a relation that assigns to every element in the domain exactly one element in the range Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time In this chapter, we will explore linear functions, their graphs, and how to relate them to data 3/3 BRICS LS-96-6 T. Bra ¨ uner: Introduction to Linear Logic BRICS Basic Research in Computer Science Introduction to Linear Logic Torben Bra ¨ uner BRICS Lecture Series LS-96-6 ISSN 1395-2048 December 1996 Copyright c  1996, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Lecture Series. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK - 8000 Aarhus C Denmark Telephone:+45 8942 3360 Telefax: +45 8942 3255 Internet: BRICS@brics.dk BRICS publications are in general accessible through World Wide Web and anonymous FTP: http://www.brics.dk/ ftp://ftp.brics.dk/ This document in subdirectory LS/96/6/ Introduction to Linear Logic Torben Bra ¨ uner Torben Bra¨uner BRICS 1 Department of Computer Science University of Aarhus Ny Munkegade DK-8000 Aarhus C, Denmark 1 Basic Research In Computer Science, Centre of the Danish National Research Foundation. Preface The main concern of this report is to give an introduction to Linear Logic. For pedagogical purposes we shall also have a look at Classical Logic as well as Intuitionistic Logic. Linear Logic was introduced by J Y. Girard in 1987 and it has attracted much attention from computer scientists, as it is a logical way of coping with resources and resource control. The focus of this technical report will be on proof-theory and computational interpretation of proofs, that is, we will focus on the question of how to interpret proofs as programs and reduction (cut-elimination) as evaluation. We first introduce Classical Logic. This is the fundamental idea of the proofs-as-programs paradigm. Cut-elimination for Classical Logic is highly non-deterministic; it is shown how this can be remedied either by moving to Intuitionistic Logic or to Linear Logic. In the case on Linear Logic we consider Intuitionistic Linear Logic as well as Classical Linear Logic. Furthermore, we take a look at the Girard Translation translating Intuitionistic Logic into Intuitionistic Linear Logic. Also, we give a brief introduction to some concrete models of Intuitionistic Linear Logic. No proofs will be given except that a proof of cut-elimination for the multiplicative fragment of Classical Linear Logic is included in an appendix. Acknowledgements. Thanks for comments from the participants of the BRICS Mini-course corresponding to this technical report. The proof-rules are produced using Paul Taylor’s macros. v vi Contents Preface v 1 Classical and Intuitionistic Logic 1 1.1 Classical Logic 1 1.2 Intuitionistic Logic 5 1.3 The λ-Calculus 8 1.4 The Curry-Howard Isomorphism 12 2 Linear Logic 14 2.1 Classical Linear Logic 14 2.2 Intuitionistic Linear Logic 19 2.3 A Digression - Russell’s Paradox and Linear Logic 23 2.4 The Linear λ-Calculus 27 2.5 The Curry-Howard Isomorphism 31 2.6 The Girard Translation 32 2.7 Concrete Models 35 A Logics 40 A.1 Classical Logic 40 A.2 Intuitionistic Logic 42 A.3 Classical Linear Logic 43 A.4 Intuitionistic Linear Logic 45 B Cut-Elimination for Classical Linear Logic 46 B.1 Some Preliminary Results 46 B.2 Putting the Proof Together 52 vii viii [...]... Curry-Howard interpretations of reductions on [...]... 559 560 5 61 564 566 568 572 574 576 16 LINEAR TRANSFORMATIONS AND MATRICES 16 .1 Linear transformations 16 .2 Nul1 space and range 16 .3 Nullity and rank 16 .4 Exercises 16 .5 Algebraic operations on linear transformations 16 .6 Inverses 16 .7 One -to- one linear transformations 16 .8 Exercises 16 .9 Linear transformations with prescribed values 16 .10 Matrix representations of linear transformations 16 .11 Construction... a matrix representation in diagonal form 16 .12 Exercises 16 .13 Linear spaces of matrices 16 .14 Isomorphism between linear transformations and matrices 16 .15 Multiplication of matrices 16 .16 Exercises 16 .17 Systems of linear equations 578 579 5 81 582 583 585 587 589 590 5 91 594 596 597 599 600 603 605 xx Contents 16 .18 Computation techniques of 16 .19 Inverses squarematrices 16 .20 Exercises 16 . 21 Miscellaneous... exponential and trigonometric functions *Il 12 Bernstein’s theorem 11 .13 Exercises 11 .14 Power series and differential equations 11 .15 The binomial series 11 .16 Exercises 435 437 438 439 4 41 443 12 VECTOR ALGEBRA 12 .1 Historical introduction 12 .2 The vector space of n-tuples of real numbers 12 .3 Geometric interpretation for n < 3 12 .4 Exercises 12 .5 The dot product 12 .6 Length or norm of a vector 12 .7 Orthogonality... ALGEBRA TO ANALYTIC GEOMETRY 13 .1 13.2 13 .3 13 .4 13 .5 13 .6 13 .7 13 .8 13 .9 Introduction Lines in n-space Some simple properties of straight lines Lines and vector-valued functions Exercises Planes in Euclidean n-space Planes and vector-valued functions Exercises The cross product 4 71 472 473 474 477 478 4 81 482 483 xv 111 Contents 13 .10 The cross product expressed as a determinant 13 .11 Exercises 13 .12 ... of vectors 12 .8 Exercises 12 .9 Projections Angle between vectors in n-space 12 .10 The unit coordinate vectors 12 .11 Exercises 12 .12 The linear span of a finite set of vectors 12 .13 Linear independence 12 .14 Bases 12 .15 Exercises 12 .16 The vector space V,(C) of n-tuples of complex 12 .17 Exercises numbers 445 446 448 450 4 51 453 455 456 457 458 460 462 463 466 467 468 470 13 APPLICATIONS OF VECTOR ALGEBRA. .. sets in a linear space 15 .8 Bases and dimension 15 .9 Exercises 15 .10 Inner products, Euclidean norms spaces, 15 .11 Orthogonality in a Euclidean space 15 .12 Exercises 15 .13 Construction of orthogonal sets The Gram-Schmidt process 15 .14 Orthogonal complements Projections 15 .15 Best approximation of elements in a Euclidean space by elements in a finitedimensional subspace 15 .16 Exercises 5 51 5 51 552 554... product 13 .13 Cramer’s rule for solving a system of three linear equations 13 .14 Exercises 13 .15 Normal vectors to planes Telemark University College Department of Electrical Engineering, Information Technology and Cybernetics Faculty of Technology, Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: +47 35 57 50 00 Fax: +47 35 57 54 01 Introduction to Visual Studio and C# HANS-PETTER HALVORSEN, 2012.08.17 2 Table of Contents 1 Introduction 5 1.1 Visual Studio 5 1.2 C# . 6 1.3 .NET Framework 6 1.4 Object-Oriented Programming (OOP) . 7 2 Visual Studio . 8 2.1 Introduction . Introduction to Linear Momentum and Collisions Introduction to Linear Momentum and Collisions Bởi: OpenStaxCollege Each rugby player has great momentum, which will affect the outcome of their collisions with each other and the ground (credit: ozzzie, Flickr) We use the term momentum in various ways in everyday language, and most of these ways are consistent with its precise scientific definition We speak of sports teams or politicians gaining and maintaining the momentum to win We also recognize that momentum has something to with collisions For example, looking at the rugby players in the photograph colliding and falling to the ground, we expect their momenta to have great effects in the resulting collisions Generally, momentum implies 1/2 Introduction to Linear Momentum and Collisions a tendency to continue on course—to move in the same direction—and is associated with great mass and speed Momentum, like energy, is important because it is conserved Only a few physical quantities are conserved in nature, and studying them yields fundamental insight into how nature works, as we shall see in our study of momentum 2/2 Embedded Systems Design: An Introduction to Processes, Tools, and Techniques by Arnold S. Berger ISBN: 1578200733 CMP Books © 2002 (237 pages) An easy-to-understand guidebook for those embarking upon an embedded processor development project. Table of Contents Embedded Systems Design—An Introduction to Processes, Tools, and Techniques Preface Introduction Chapter 1 - The Embedded Design Life Cycle Chapter 2 - The Selection Process Chapter 3 - The Partitioning Decision Chapter 4 - The Development Environment Chapter 5 - Special Software Techniques Chapter 6 - A Basic Toolset Chapter 7 - BDM, JTAG, and Nexus Chapter 8 - The ICE — An Integrated Solution Chapter 9 - Testing Chapter 10 - The Future Index List of Figures List of Tables List of Listings List of Sidebars TEAMFLY Team-Fly ® Embedded Systems Design—An Introduction to Processes, Tools, and Techniques Arnold Berger CMP Books CMP Media LLC 1601 West 23rd Street, Suite 200 Lawrence, Kansas 66046 USA www.cmpbooks.com Designations used by companies to distinguish their products are often claimed as trademarks. In all instances where CMP Books is aware of a trademark claim, the product name appears in initial capital letters, in all capital letters, or in accordance with the vendor’s capitalization preference. Readers should contact the appropriate companies for more complete information on trademarks and trademark registrations. All trademarks and registered trademarks in this book are the property of their respective holders. Copyright © 2002 by CMP Books, except where noted otherwise. ... of bamboo has been observed to grow nearly 1.5 inches every hour http://www.guinnessworldrecords.com/records-3000/fastest-growing-plant/ 2/3 Introduction to Linear Functions In a twenty-four hour... plant, is a linear function Recall from Functions and Function Notation that a function is a relation that assigns to every element in the domain exactly one element in the range Linear functions. .. that can be used to model many real-world applications, such as plant growth over time In this chapter, we will explore linear functions, their graphs, and how to relate them to data 3/3