There are many reasons why neural networks fascinate us and have captivated headlines in recent years. They make web searches better, organize photos, and are even used in speech translation. Heck, they can even generate encryption. At the same time, they are also mysterious and mind-bending: how exactly do they accomplish these things ? What goes on inside a neural network?On a high level, a network learns just like we do, through trial and error. This is true regardless if the network is supervised, unsupervised, or semi-supervised. Once we dig a bit deeper though, we discover that a handful of mathematical functions play a major role in the trial and error process. It also becomes clear that a grasp of the underlying mathematics helps clarify how a network learns. In the following chapters we will unpack the mathematics that drive a neural network. To do this, we will use a feedforward network as our model and follow input as it moves through the network. Read more

Are you ready to dive into the vibrant world of linear algebra and see how it powers real-world applications? Welcome to this comprehensive guide, where traditional theory meets modern computational practices.Linear algebra is the magic behind many computational sciences - machine learning, AI, data science, statistics, simulations, computer graphics, multivariate analyses, matrix decompositions, signal processing, and more. But here's a secret: the way it's taught in traditional textbooks isn't how professionals use it in the field.For instance, have you ever wondered about the practical importance of a matrix's "determinant"? You might be in for a surprise! This book bridges the gap between theoretical understanding and practical application, showing you not only the 'what' but also the 'how' of implementing linear algebra in real-world scenarios.What makes this book a must-have resource?Crystal clear explanations of linear algebra concepts and theories.Multiple angles to explain ideas, a proven technique to help cement your understanding.Vivid graphical visualizations to enhance your geometric intuition of linear algebra.Real-world implementations in MATLAB and Python. After all, in today's world, you seldom solve math problems by hand. Software is the way forward!A range of topics from beginner to intermediate levels, including vectors, matrix multiplications, least-squares projections, eigendecomposition, and singular-value decomposition.Emphasis on the application-oriented aspects of linear algebra and matrix analysis.Intuitive visual explanations of diagonalization, eigenvalues and eigenvectors, and singular value decomposition.Ready-to-use codes in MATLAB and Python to bring linear algebra concepts to life on your computer. All codes can be downloaded from https://github.com/mikexcohen/LinAlgBook.A unique blend of hand-solved exercises and advanced code challenges. Remember, math is not a spectator sport!Whether you're just starting your journey in linear algebra or seeking to apply these concepts to data analyses on computers (such as statistics or signal processing), this book is your go-to guide. With this book at your side, you won't just learn linear algebra; you'll experience it! Read more

An engaging introduction to vectors and matrices and the algorithms that operate on them, intended for the student who knows how to program. Mathematical concepts and computational problems are motivated by applications in computer science. The reader learns by doing, writing programs to implement the mathematical concepts and using them to carry out tasks and explore the applications. Examples include: error-correcting codes, transformations in graphics, face detection, encryption and secret-sharing, integer factoring, removing perspective from an image, PageRank (Google's ranking algorithm), and cancer detection from cell features. A companion web site, codingthematrix.com, provides data and support code. Most of the assignments can be auto-graded online. Over two hundred illustrations, including a selection of relevant xkcd comics. Read more

Precalculus for Legends: Synthetic Division, Exponentials, Logarithms, Matrices, Partial Fractions, and SequencesAre you ready to take action and master essential precalculus topics? This book is inspired by the principles of Real Superpowers That Will Change Your Life and is part of the Superpowered Mathematics Series, designed to empower you to dive into math with confidence and energy.This fun and approachable guide focuses on key topics often encountered in algebra and precalculus courses, helping you build specific skills that will prepare you for calculus and beyond.What You’ll Find in This Book:Synthetic Division: A straightforward way to divide polynomials.Exponentials and Logarithms: Master exponentials and logarithms through practice.Matrices: Understand the power of arrays and how to manipulate them.Partial Fractions: Learn to decompose complex fractions into manageable parts.Sequences: Dive into patterns and progressions to sharpen your problem-solving skills.Why You’ll Love This Book:Action-Oriented: Inspired by Real Superpowers That Will Change Your Life, this book focuses on action, helping you get hands-on with math.Concise and Focused: A quick, enjoyable read with varied topics that keep things fresh and engaging.Complete Solutions: Full solutions to problems are included, so you can practice with confidence.If you’re ready to stop overthinking and start doing, this book is your gateway to mathematical success.Let’s GO! Take action now and become a math legend! Read more

Demystify quantum computing by learning the math it is built onKey FeaturesBuild a solid mathematical foundation to get started with developing powerful quantum solutionsUnderstand linear algebra, calculus, matrices, complex numbers, vector spaces, and other concepts essential for quantum computingLearn the math needed to understand how quantum algorithms functionBook DescriptionQuantum computing is an exciting subject that offers hope to solve the world's most complex problems at a quicker pace. It is being used quite widely in different spheres of technology, including cybersecurity, finance, and many more, but its concepts, such as superposition, are often misunderstood because engineers may not know the math to understand them. This book will teach the requisite math concepts in an intuitive way and connect them to principles in quantum computing.Starting with the most basic of concepts, 2D vectors that are just line segments in space, you'll move on to tackle matrix multiplication using an instinctive method. Linearity is the major theme throughout the book and since quantum mechanics is a linear theory, you'll see how they go hand in hand. As you advance, you'll understand intrinsically what a vector is and how to transform vectors with matrices and operators. You'll also see how complex numbers make their voices heard and understand the probability behind it all.It's all here, in writing you can understand. This is not a stuffy math book with definitions, axioms, theorems, and so on. This book meets you where you're at and guides you to where you need to be for quantum computing. Already know some of this stuff? No problem! The book is componentized, so you can learn just the parts you want. And with tons of exercises and their answers, you'll get all the practice you need.What you will learnOperate on vectors (qubits) with matrices (gates)Define linear combinations and linear independenceUnderstand vector spaces and their basis setsRotate, reflect, and project vectors with matricesRealize the connection between complex numbers and the Bloch sphereDetermine whether a matrix is invertible and find its eigenvaluesProbabilistically determine the measurement of a qubitTie it all together with bra-ket notationWho this book is forIf you want to learn quantum computing but are unsure of the math involved, this book is for you. If you've taken high school math, you'll easily understand the topics covered. And even if you haven't, the book will give you a refresher on topics such as trigonometry, matrices, and vectors. This book will help you gain the confidence to fully understand quantum computation without losing you in the process!Table of ContentsSuperposition with EuclidThe MatrixFoundationsVector SpacesUsing Matrices to Transform SpaceComplex NumbersEigenstuffOur Space in the UniverseAdvanced ConceptsAppendix 1 - Bra-ket NotationAppendix 2 - Sigma NotationAppendix 3 - TrigonometryAppendix 4 - ProbabilityAppendix 5 - References Read more

Advanced undergraduate and first-year graduate students have long regarded this text as one of the best available works on matrix theory in the context of modern algebra. Teachers and students will find it particularly suited to bridging the gap between ordinary undergraduate mathematics and completely abstract mathematics.The first five chapters treat topics important to economics, psychology, statistics, physics, and mathematics. Subjects include equivalence relations for matrixes, postulational approaches to determinants, and bilinear, quadratic, and Hermitian forms in their natural settings. The final chapters apply chiefly to students of engineering, physics, and advanced mathematics. They explore groups and rings, canonical forms for matrixes with respect to similarity via representations of linear transformations, and unitary and Euclidean vector spaces. Numerous examples appear throughout the text. Read more

This concise, plain-language guide for senior undergraduates and graduate students aims to develop intuition, practical skills and an understanding of the framework of numerical methods for the physical sciences and engineering. It provides accessible self-contained explanations of mathematical principles, avoiding intimidating formal proofs. Worked examples and targeted exercises enable the student to master the realities of using numerical techniques for common needs such as solution of ordinary and partial differential equations, fitting experimental data, and simulation using particle and Monte Carlo methods. Topics are carefully selected and structured to build understanding, and illustrate key principles such as: accuracy, stability, order of convergence, iterative refinement, and computational effort estimation. Enrichment sections and in-depth footnotes form a springboard to more advanced material and provide additional background. Whether used for self-study, or as the basis of an accelerated introductory class, this compact textbook provides a thorough grounding in computational physics and engineering. Read more

Algebra II For Dummies, 2nd Edition (9781119543145) was previously published as Algebra II For Dummies, 2nd Edition (9781119090625). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.  Your complete guide to acing Algebra II Do quadratic equations make you queasy? Does the mere thought of logarithms make you feel lethargic? You're not alone! Algebra can induce anxiety in the best of us, especially for the masses that have never counted math as their forte. But here's the good news: you no longer have to suffer through statistics, sequences, and series alone. Algebra II For Dummies takes the fear out of this math course and gives you easy-to-follow, friendly guidance on everything you'll encounter in the classroom and arms you with the skills and confidence you need to score high at exam time.Gone are the days that Algebra II is a subject that only the serious 'math' students need to worry about. Now, as the concepts and material covered in a typical Algebra II course are consistently popping up on standardized tests like the SAT and ACT, the demand for advanced guidance on this subject has never been more urgent. Thankfully, this new edition of Algebra II For Dummies answers the call with a friendly and accessible approach to this often-intimidating subject, offering you a closer look at exponentials, graphing inequalities, and other topics in a way you can understand. Examine exponentials like a proFind out how to graph inequalitiesGo beyond your Algebra I knowledgeAce your Algebra II exams with easeWhether you're looking to increase your score on a standardized test or simply succeed in your Algebra II course, this friendly guide makes it possible.  Read more

Computers can't LEARN... Right?!Machine Learning is a branch of computer science that wants to stop programming computers using a detailed list of commands to follow blindly. Instead, their aim is to implement high-level routines that teach computers how to approach new and unknown problems – these are called algorithms.In practice, they want to give computers the ability to Learn and to Adapt.We can use these algorithms to obtain insights, recognize patterns and make predictions from data, images, sounds or videos we have never seen before – or even knew existed. Unfortunately, the true power and applications of today’s Machine Learning Algorithms remain deeply misunderstood by most people.Through this book I want fix this confusion, I want to shed light on the most relevant Machine Learning Algorithms used in the industry. I will show you exactly how each algorithm works, why it works and when you should use it.Supervised Learning AlgorithmsK-Nearest NeighbourNaïve BayesRegressionsUnsupervised Learning Algorithms:Support Vector MachinesNeural NetworksDecision Trees Read more

Written as a guide to using matrices as a mathematical tool, this text is geared toward physical and social scientists, engineers, economists, and others who require a model for procedure rather than an exposition of theory. Knowledge of elementary algebra is the only mathematical prerequisite. Detailed numerical examples illustrate the treatment's focus on computational methods. The first four chapters outline the basic concepts of matrix theory. Topics include the development of the concept of elementary operations and a systematic procedure for simplifying matrices as well as a method for evaluating the determinant of a given square matrix. Subsequent chapters explore important numerical procedures, including the process for approximating characteristic roots and vectors plus direct and iterative methods for inverting matrices and solving systems of equations. Solutions to the problems are included. Read more

"This is a book for beginners in linear algebra. Duc Tran aims for simple statements of the ideas and facts that he needs. He organizes them in a coherent way: first the main facts about vectors and then about matrices. For a matrix, he discusses elimination to echelon form and then the rank and the four fundamental subspaces. The result is a short book that a student can read." - Professor Gilbert Strang, Massachusetts Institute of Technology (MIT)"The book adopts a visual and largely informal approach to introduce linear algebra. It reverses the old-school order which started from linear equations or operation on matrices; instead, the author begins with the graphical representation of vectors, whilst the determinants and linear equations are postponed to the last two chapters. He introduces the notion of norms at an early stage, including the l^1 or 'taxicab' norm, which is not only interesting but also indispensable for advanced topics. The appendices contain rough sketches of advanced topics such as tensors. This book will be a nice complement to the more comprehensive textbooks on linear algebra. It will be welcomed by students who feel frustrated by the orthodox pedagogy to this topic." - Professor Wen-Wei Li, Peking University"This introductory level textbook for basic linear algebra written by Duc Tran is for those who wish to study linear algebra for the first time. It will be especially suitable for high school students or first year undergraduates when they have to understand basic linear algebra but do not want to go to the heavy mathematics course. The author provides plenty of graphs and graphical representations of matrices with examples, exercises and their solutions. This book reminds the reviewer some images like AP course, self-study, linear algebra for biology or social science majors, and linear algebra for programmers." - Professor Sang Geun Han, Korea Advanced Institute of Science and Technology (KAIST) Read more

This emphasis on this book is on the methods and techniques of Linear Algebra, as well as few number of fundamental ideas and the relationships among them.In my experience, learning by actually doing is the best way.You can't learn by watching someone else doing it! The material in each chapter can (and should) be finished inone week. There is a sufficient number of problems withsolutions. The explanations are as short as possible, but notshorter. The exercises are ordered from easy to difficult. Whilethe number of questions is not very large, almost every trick ofthe trade is covered by one of them. Read more

Linear algebra has broad application in physics, business, data science, and engineering. This book offers a clear and concise introduction to its key concepts, without getting into the weeds of the subject. The emphasis is on understanding what linear algebra is about and how matrices and vectors work. Here's what makes this book accessible:Visual Learning: Numerous figures illustrate mathematical equations.Focus on Clarity: Complicated ideas are broken down into smaller, intuitively clear parts.Smooth Reading Flow: Lengthy derivations are placed in appendices to maintain a clear narrative.No prerequisites are assumed beyond high school level math. The book starts by discussing linear operators, vectors, and a visual understanding of what an operator can do when applied to a vector. Then the book proceeds to the standard linear algebra material: matrices, solving linear equations, eigenvectors, and more. One chapter is dedicated to a discussion of several major linear algebra applications; in addition, references to many other applications are scattered throughout the text. Read more

Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on the subject by exploring the connections of these tools to matrices. Placing combinatorial and graph-theoretical tools at the forefront of the development of matrix theory, this book uses graphs to explain basic matrix construction, formulas, computations, ideas, and results. It presents material rarely found in other books at this level, including Gersgorin's theorem and its extensions, the Kronecker product of matrices, sign-nonsingular matrices, and the evaluation of the permanent matrix. The authors provide a combinatorial argument for the classical Cayley-Hamilton theorem and a combinatorial proof of the Jordan canonical form of a matrix. They also describe several applications of matrices in electrical engineering, physics, and chemistry. Read more

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Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related topics such as determinants, eigenvalues, and differential equations.Table of Contents:l. The Algebra of Matrices2. Linear Equations3. Vector Spaces4. Determinants5. Linear Transformations6. Eigenvalues and Eigenvectors7. Inner Product Spaces8. Applications to Differential EquationsFor the second edition, the authors added several exercises in each chapter and a brand new section in Chapter 7. The exercises, which are both true-false and multiple-choice, will enable the student to test his grasp of the definitions and theorems in the chapter. The new section in Chapter 7 illustrates the geometric content of Sylvester's Theorem by means of conic sections and quadric surfaces. 6 line drawings. lndex. Two prefaces. Answer section. Read more

El álgebra es una de las disciplinas matemáticas más importantes y con amplias aplicaciones. Este libro ofrece una introducción sólida al fascinante mundo del álgebra desde los conceptos más básicos hasta temas avanzados.Contenido del LibroExploración de números enteros, notación algebraica y propiedades fundamentales.Ecuaciones, desigualdades, sistemas y su representación gráfica.Polinomios, factorización, funciones y sus gráficos.Ecuaciones cuadráticas, funciones cuadráticas, conjuntos y álgebra booleana.Proporciones, exponentes, radicales y funciones exponenciales-logarítmicas.Ejercicios para reforzar conceptos y ejemplos prácticos de aplicaciones en la vida cotidiana.Un capítulo final que explora los fascinantes números complejos.Este texto proporciona una sólida base en álgebra que prepara al lector para resolver problemas matemáticos avanzados y comprender aplicaciones en diversas disciplinas. ¡Una herramienta valiosa para estudiantes y cualquier persona interesada en las matemáticas! Read more

Linear Models and the Relevant Distributions and Matrix Algebra provides in-depth and detailed coverage of the use of linear statistical models as a basis for parametric and predictive inference. It can be a valuable reference, a primary or secondary text in a graduate-level course on linear models, or a resource used (in a course on mathematical statistics) to illustrate various theoretical concepts in the context of a relatively complex setting of great practical importance.Features:Provides coverage of matrix algebra that is extensive and relatively self-contained and does so in a meaningful contextProvides thorough coverage of the relevant statistical distributions, including spherically and elliptically symmetric distributionsIncludes extensive coverage of multiple-comparison procedures (and of simultaneous confidence intervals), including procedures for controlling the k-FWER and the FDRProvides thorough coverage (complete with detailed and highly accessible proofs) of results on the properties of various linear-model procedures, including those of least squares estimators and those of the F test.Features the use of real data sets for illustrative purposesIncludes many exercises Read more

Not only is matrix theory significant in a wide range of fields mathematical economics, quantum physics, geophysics, electrical network synthesis, crystallography, and structural engineering, among others-but with the vast proliferation of digital computers, knowledge of matrix theory is a must for every modern engineer, mathematician, and scientist. Matrices represent linear transformations from a finiteset of numbers to another finite set of numbers.Since many important problems are linear, and since digital computers with finite memory manipulate only finite sets of numbers, the solution of linear problems by digital computers usually involves matrices. Developed from the author's course on matrix theory at the CaliforniaInstitute of Technology, the book begins with a concise presentation of the theory of determinants, continues with a discussion of classical linear algebra, and an optional chapter on the use of matrices to solve systems of linear triangularizations of Hermitian and nonHermitian matrices, as well as a chapter presenting a proof of the difficult and important matrix theory of Jordan. The book concludes with discussions of variational principles and perturbation theory of matrices, matrix numerical analysis, and an introduction to the subject of linear computations.The book is designed to meet many different needs, and because it is mathematically rigorous, it may be used by students of pure and applied mathematics. Since it is oriented towards applications, it is valuable to students of engineering, science, and the social sciences. And because it contains the basic preparation in matrix theory required for numerical analysis, it can be used by students whose main interest is computers. The book assumes very little mathematical preparation, and except for the single section on the continuous dependence of eigenvalues on matrices, a knowledge of elementary algebra and calculus is sufficient. Read more

This is a comprehensive eBook that serves as a valuable resource for engineers seeking a strong mathematical foundation in their respective fields. The eBook covers a wide range of mathematical concepts and techniques that are essential for solving engineering problems and understanding complex systems.The eBook is divided into several chapters, each focusing on a specific area of mathematics applicable to engineering. It covers algebraic methods, including linear equations, matrices, polynomial equations, and exponents and logarithms, the calculus section explores differentiation, integration, and multivariable calculus differential equations delves into both first-order and second-order ordinary differential equations, covering topics such as separable equations, exact equations, and homogeneous and non-homogeneous equations, providing practical examples and solutions. This is an invaluable resource for engineers seeking to strengthen their mathematical expertise and apply it to their engineering practice. With its practical approach, comprehensive coverage, and real-world examples, this eBook serves as an indispensable guide for engineers in various disciplines. The eBook concludes with a chapter dedicated to practical questions and answers. It provides a collection of engineering problems, along with detailed solutions and explanations. Common challenges and troubleshooting tips are also shared, enabling engineers to tackle complex mathematical problems with confidence.Throughout the ebook, the importance of continuous learning in engineering mathematics is emphasized. It highlights the need for engineers to continually enhance their mathematical skills to stay abreast of advancements in their fields and solve complex engineering challenges effectively. Read more