Linear Algebra for Everyone      Gilbert Strang

ISBN 978-1-7331466-3-0               September 2020 Wellesley-Cambridge Press          [email protected]

Linear Algebra For Everyone Cover

Publication September 2020

Gilbert Strang [email protected]

This new linear algebra textbook will be available for purchase in mid-September 2020

For information please email [email protected]

Other books by Gilbert Strang Dummy

This is a textbook for everyone

Table of Contents and Preface

Solution manual : linear algebra for everyone (november 2023), new ideas in linear algebra for everyone, section 1.3             section 1.4        of  this  book, section 1.4, image compression by the svd, tim baumann's website, a 2020 vision of linear algebra (videos), lu and cr elimination (to appear in siam review), errata in the first printing, svd of real matrices, presentation by prof. jugal k. verma, iit bombay (march 2020), we have improved the explanation of how to find the nullspace of any matrix a  :,     we want all solutions to ax = 0  : all combinations of columns leading to zero vector,     there are n - r special solutions after elimination simplifies the equations,     new version of section 3.2, in the next printing of linear algebra for everyone, new 3.2 : a = cr and computing the nullspace by elimination, original 3.2 : the nullspace of a : solving ax = 0.

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Wellesley-Cambridge Press           : All Gilbert Strang's textbooks

American Mathematical Society  : Distributed within the Americas

Cambridge University Press          : Outside North America

Kyobo Books and Ting Lung          : Books in Korea and Taiwan

Wellesley Publishers (India)         : Indian editions of select books

Review of the 5th edition by Professor Farenick for the International Linear Algebra Society

Related websites: ocw.mit.edu , math.mit.edu/~gs

ISBN: 978-1-7331466-3-0

[Table of Contents]

Practice Exam Questions

Table of contents for introduction to linear algebra (5th edition 2016).

  • 1.1 Vectors and Linear Combinations
  • 1.2 Lengths and Dot Products
  • 1.3 Matrices
  • 2.1 Vectors and Linear Equations
  • 2.2 The Idea of Elimination
  • 2.3 Elimination Using Matrices
  • 2.4 Rules for Matrix Operations
  • 2.5 Inverse Matrices
  • 2.6 Elimination = Factorization: A = LU
  • 2.7 Transposes and Permutations
  • 3.1 Spaces of Vectors
  • 3.2 The Nullspace of A : Solving Ax = 0 and Rx = 0
  • 3.3 The Complete Solution to Ax = b
  • 3.4 Independence, Basis and Dimension
  • 3.5 Dimensions of the Four Subspaces
  • 4.1 Orthogonality of the Four Subspaces
  • 4.2 Projections
  • 4.3 Least Squares Approximations
  • 4.4 Orthonormal Bases and Gram-Schmidt
  • 5.1 The Properties of Determinants
  • 5.2 Permutations and Cofactors
  • 5.3 Cramer’s Rule, Inverses, and Volumes
  • 6.1 Introduction to Eigenvalues
  • 6.2 Diagonalizing a Matrix
  • 6.3 Systems of Differential Equations
  • 6.4 Symmetric Matrices
  • 6.5 Positive Definite Matrices
  • 7.1 Image Processing by Linear Algebra
  • 7.2 Bases and Matrices in the SVD
  • 7.3 Principal Component Analysis (PCA by the SVD)
  • 7.4 The Geometry of the SVD
  • 8.1 The Idea of a Linear Transformation
  • 8.2 The Matrix of a Linear Transformation
  • 8.3 The Search for a Good Basis
  • 9.1 Complex Numbers
  • 9.2 Hermitian and Unitary Matrices
  • 9.3 The Fast Fourier Transform
  • 10.1 Graphs and Networks
  • 10.2 Matrices in Engineering
  • 10.3 Markov Matrices, Population, and Economics
  • 10.4 Linear Programming
  • 10.5 Fourier Series: Linear Algebra for Functions
  • 10.6 Computer Graphics
  • 10.7 Linear Algebra for Cryptography
  • 11.1 Gaussian Elimination in Practice
  • 11.2 Norms and Condition Numbers
  • 11.3 Iterative Methods and Preconditioners
  • 12.1 Mean, Variance, and Probability
  • 12.2 Covariance Matrices and Joint Probabilities
  • 12.3 Multivariate Gaussian andWeighted Least Squares
  • Matrix Factorizations
  • Six Great Theorems / Linear Algebra in a Nutshell

Lecture Notes for Linear Algebra (ebook, 2021)

Buy the ebook from google playstore, the art of linear algebra, by kenji hiranabe, interesting links, matrix world : the picture of all matrices, by kenji hiranabe (appears in this book), other books by gilbert strang, introduction to linear algebra, linear algebra and learning from data, differential equations and linear algebra, computational science and engineering.

Each section of the book has a Problem Set.

Introduction to Linear Algebra, International Edition Click here to download Selected Solutions

Links to websites for each semester at MIT:   web.mit.edu/18.06 ,

  • Exam 1 (1997-2009)
  • Exam 1 (2010-2015)
  • Exam 2 (1997-2009)
  • Exam 2 (2010-2015)
  • Exam 3 (1997-2009)
  • Exam 3 (2010-2015)
  • Final (1998-2009)
  • Final (2010-2015)

Linear Algebra Problems in Lemma

My friend Pavel Grinfeld at Drexel has sent me a collection of interesting problems -- mostly elementary but each one with a small twist. These are part of his larger teaching site called LEM.MA and he built the page http://lem.ma/LAProb /especially for this website linked to the 5th edition.

The H.264 Video Standard (promised in Section 7.1 of the book)

This video standard describes a system for encoding and decoding (a "Codec") that engineers have defined for applications like High Definition TV. It is not expected that you will know the meaning of every word -- your book author does not know either. The point is to see an important example of a "standard" that is created by an industry after years of development--- so all companies will know what coding system their products must be consistent with.

The words "motion compensation" refer to a way to estimate each video image from the previous one. The simplest would be to guess that successive video images are the same. Then we would only need the changes between frames -- hopefully small. But if the camera is following the action, the whole scene will shift slightly and need correction. A better idea is to see which way the scene is moving and build that change into the next scene. This is MOTION COMPENSATION. In fact the motion is allowed to be different on different parts of the screen.

It is ideas like this -- easy to talk about but taking years of effort to perfect -- that make video technology and other technologies possible and successful. Engineers do their job. I hope these links give an idea of the detail needed.

  • http://www.h264info.com/h264.html
  • http://en.wikipedia.org/wiki/H.264/MPEG-4_AVC
  • http://www.axis.com/files/whitepaper/wp_h264_31669_en_0803_lo.pdf

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In Algebra 1, students engage with various expressions from an algebraic and geometric perspective. They learn to solve linear and quadratic equations as well as how to represent expressions in the Cartesian plane. Students are imbued with a deep understanding of the general concept behind functions and are introduced to several important types of functions. Students are also introduced to the rich field of complex numbers. As with previous courses, we continue to emphasize challenging word problems to train students how to apply their new tools. Advanced problems from contests such as MATHCOUNTS, AMC 8, and AMC 10 are included throughout the course.

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Students should expect to spend 1-3 hours on homework every week. Homework will be assigned and completed through the class website, where students receive immediate feedback on their work. Students will take three in-class exams throughout the course.

The textbook for the course is Introduction to Algebra by Art of Problem Solving.

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Learn the basics of algebra from former USA Mathematical Olympiad winner and Art of Problem Solving founder Richard Rusczyk. Topics covered in the book include linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value, sequences and series, and much more!

The text is structured to inspire the reader to explore and develop new ideas. Each section starts with problems, giving the student a chance to solve them without help before proceeding. The text then includes solutions to these problems, through which algebraic techniques are taught. Important facts and powerful problem solving approaches are highlighted throughout the text. In addition to the instructional material, the book contains well over 1000 problems. The solutions manual contains full solutions to all of the problems, not just answers.

This book can serve as a complete Algebra I course, and also includes many concepts covered in Algebra II. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of algebra will find this book an instrumental part of their mathematics libraries.

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Introduction To Algebra (the Art Of Problem Solving) By Richard Rusczyk

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Richard Rusczyk

Introduction to Algebra 2nd Edition

  • ISBN-10 1934124141
  • ISBN-13 978-1934124147
  • Edition 2nd
  • Publisher AoPS Incorporated
  • Publication date March 30, 2007
  • Part of series Art of Problem Solving
  • Language English
  • Dimensions 8.75 x 1.5 x 11.25 inches
  • Print length 656 pages
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Art of Problem Solving Introduction to Algebra Textbook and Solutions Manual 2-Book Set

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  • Publisher ‏ : ‎ AoPS Incorporated; 2nd edition (March 30, 2007)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 656 pages
  • ISBN-10 ‏ : ‎ 1934124141
  • ISBN-13 ‏ : ‎ 978-1934124147
  • Item Weight ‏ : ‎ 1.95 pounds
  • Dimensions ‏ : ‎ 8.75 x 1.5 x 11.25 inches
  • #459 in Algebra
  • #1,217 in Decision-Making & Problem Solving
  • #12,559 in Unknown

About the author

Richard rusczyk.

Richard Rusczyk founded Art of Problem Solving (AoPS) in 2003 to create interactive educational opportunities for avid math students. Richard is one of the co-authors of the Art of Problem Solving classic textbooks, author of Art of Problem Solving's Introduction to Algebra, Introduction to Geometry, and Precalculus textbooks, co-author of Art of Problem Solving's Intermediate Algebra and Prealgebra, one of the co-creators of the Mandelbrot Competition, and a past Director of the USA Mathematical Talent Search. He was a participant in National MATHCOUNTS, a three-time participant in the Math Olympiad Summer Program, and a USA Mathematical Olympiad winner (1989). He graduated from Princeton University in 1993, and worked as a bond trader for D.E. Shaw & Company for four years. AoPS marks Richard's return to his vocation: educating motivated students.

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FindUrTalent

Introduction to Linear Algebra

Linear algebra is a foundational branch of mathematics that deals with vector spaces, linear transformations, and systems of linear equations. It provides essential tools and techniques for solving a wide range of problems in mathematics, science, engineering, and computer science. Understanding linear algebra is crucial for grasping advanced concepts in areas such as machine learning, quantum mechanics, computer graphics, and optimization.

Basic Concepts:1. Vectors and Vector Spaces: – Vector: A vector is a mathematical object that represents a quantity with both magnitude and direction. Vectors can be represented geometrically as arrows in space or algebraically as ordered lists of numbers. – Vector Space: A vector space is a set of vectors equipped with operations of addition and scalar multiplication that satisfy certain properties, such as closure, associativity, commutativity, and distributivity.

2. Matrices and Linear Transformations: – Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. Matrices can represent linear transformations, systems of linear equations, and various mathematical operations. – Linear Transformation: A linear transformation is a mapping between vector spaces that preserves vector addition and scalar multiplication. It can be represented by a matrix and encompasses concepts such as rotations, scaling, reflections, and projections.

3. Systems of Linear Equations: – System of Linear Equations: A system of linear equations consists of multiple linear equations involving the same set of variables. Solutions to such systems correspond to points of intersection (if any) in multidimensional space. – Gaussian Elimination: Gaussian elimination is a systematic method for solving systems of linear equations by performing elementary row operations to transform the system into an equivalent, simpler form.4. Eigenvalues and Eigenvectors: – Eigenvalue and Eigenvector: An eigenvalue of a square matrix is a scalar λ such that there exists a nonzero vector v, called the eigenvector, satisfying the equation Av = λv, where A is the matrix. Eigenvalues and eigenvectors play a crucial role in understanding the behavior of linear transformations and systems of differential equations.

Examples:1. Vector Addition and Scalar Multiplication: – Vector Addition: Given vectors v = [2, 3] and w = [1, -1], the sum v + w = [2+1, 3+(-1)] = [3, 2]. – Scalar Multiplication: Given vector v = [2, 3] and scalar k = 2, the scalar product kv = 2[2, 3] = [4, 6].2. Matrix Multiplication: – Matrix A = [[1, 2], [3, 4]] and Matrix B = [[5, 6], [7, 8]]. – The product AB = [[1*5 + 2*7, 1*6 + 2*8], [3*5 + 4*7, 3*6 + 4*8]] = [[19, 22], [43, 50]].

Applications:Linear algebra has diverse applications across various fields, including:- Computer graphics: Transformations, rendering, and image processing.- Machine learning: Dimensionality reduction, regression, and neural networks.- Quantum mechanics: Quantum states, operators, and observables.- Economics: Input-output models, optimization, and game theory.- Signal processing: Filtering, compression, and noise reduction.- Robotics: Kinematics, dynamics, and control theory.

Conclusion:Linear algebra serves as a fundamental toolkit for modeling, analyzing, and solving problems in numerous disciplines. Its concepts and techniques provide a versatile framework for understanding complex systems, making predictions, and optimizing processes. Mastery of linear algebra is essential for advancing in fields that rely heavily on mathematical modeling and computational methods, making it a cornerstone of modern mathematics and its applications.

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  1. Art of Problem Solving: Linear Equations in Disguise Part 2

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  2. Art of Problem Solving: Solving Linear Equations Part 1

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  3. Art of Problem Solving: Linear Equations in Disguise Part 1

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  4. Art of Problem Solving: Linear Equations Special Cases

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  5. Art of Problem Solving: Solving Linear Equations Part 2

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  6. Art of Problem Solving: Simplifying Linear Expressions

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COMMENTS

  1. Art of Problem Solving

    Linear algebra - AoPS Wiki. Art of Problem Solving. AoPS Online. Math texts, online classes, and more. for students in grades 5-12. Visit AoPS Online ‚. Books for Grades 5-12 Online Courses. Beast Academy. Engaging math books and online learning.

  2. Art of Problem Solving: Simplifying Linear Expressions

    Art of Problem Solving's Richard Rusczyk explains how to simplify one-variable expressions.

  3. Art of Problem Solving: Solving Systems of Linear Equations with

    Art of Problem Solving's Richard Rusczyk solves two-variable systems of linear equations with substitution.

  4. Linear Algebra for Everyone, Gilbert Strang

    Solution Manual : Linear Algebra for Everyone (November 2023) New ideas in Linear Algebra for Everyone; Section 1.3 Section 1.4 of this book. Image compression by the SVD, Tim Baumann's website; A 2020 Vision of Linear Algebra (videos) LU and CR Elimination (to appear in SIAM Review) Errata in the First Printing

  5. Art of Problem Solving: Solving Linear Equations Part 1

    Art of Problem Solving's Richard Rusczyk solves some basic linear equations.

  6. PDF Introduction to Algebra

    Art of Problem Solving, v associative property of addition, 9 associative property of multiplication, 10 asymptote, 576 axis of symmetry, 405 base, 28, 546 ... linear algebra, 155 linear equations graphs of, 208-245 multi-variable, 123-150 one-variable, 77-98 two-variable, 123-145 linear programming, 274-279

  7. Introduction to Algebra Textbook

    Introduction to Algebra. Richard Rusczyk. Paperback (2nd edition) Text: 656 pages. Solutions: 312 pages. A thorough introduction for students in grades 6-9 to algebra topics such as linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities ...

  8. Algebra 1

    Students in Algebra 1 learn to manipulate a variety of expressions algebraically and geometrically, practice solving linear and quadratic equations, and develop familiarity with representing various expressions in the Cartesian plane. We also introduce students to the rich field of complex numbers and to important functions and concepts in discrete math.

  9. Art of Problem Solving: Graphing Linear Inequalities

    Art of Problem Solving's Richard Rusczyk explains how to graph a two-variable linear inequality on the Cartesian plane.

  10. I am considering buying Art of Problem Solving. What's your ...

    As you know, Art of Problem Solving includes 11 books that comes with their solutions and they are PreAlgebra, Introduction to Algebra, Introduction to Counting and Probability, Introduction to Geometry, Introduction to Number Theory, Intermediate Algebra, Intermediate Counting and Probability, Precalculus, Calculus, Problem Solving Volume 1: The Basics, Problem Solving Volume 2: And Beyond

  11. PDF 5.2 Solving Linear Equations I

    5.2. SOLVING LINEAR EQUATIONS I Problem 5.8: Solve the equation x12 = 289. Solution for Problem 5.8: We present three di↵erent solutions. Inspection. The equation means that 12 less than x equals 289. Since 289 is 12 less than x, we know that x must be 12 more than 289. Therefore, x equals 289+12, which is 301. Number Line.

  12. Art of Problem Solving Introduction to Algebra

    Learn the basics of algebra from former USA Mathematical Olympiad winner and Art of Problem Solving founder Richard Rusczyk. Topics covered in the book include linear equations, ratios, quadratic equations, special factorizations, complex numbers, graphing linear and quadratic equations, linear and quadratic inequalities, functions, polynomials, exponents and logarithms, absolute value ...

  13. PDF Art of Problem Solving Pre-Test Introduction to Algebra

    Art of Problem Solving Pre-Test Introduction to Algebra 3. Linear Equations. Solve each of the following equations: (a)3r 4 = 16 7r (b) 2x 3 5 = 4 3x 7 (c)2 t 4 = 3 5 t 6 4. Exponent Laws. Express each of the following as a power of 2: (a)27 26 (b) 213 25 (c)(24)3 (d)23 27 22 3 5. Ratio and Rates. (a)The ratio of boys to girls at a summer camp ...

  14. Art of Problem Solving: Linear Equations Special Cases

    Art of Problem Solving's Richard Rusczyk examines a couple special cases of linear equations.

  15. Art of Problem Solving

    Art of Problem Solving offers two other multifaceted programs. Beast Academy is our comic-based online math curriculum for students ages 6-13. And AoPS Academy brings our methodology to students grades 2-12 through small, in-person classes at local campuses. Through our three programs, AoPS offers the most comprehensive honors math pathway ...

  16. Introduction To Algebra (the Art Of Problem Solving) By Richard Rusczyk

    Introduction To Algebra (the Art Of Problem Solving) By Richard Rusczyk Addeddate 2023-07-27 11:04:53 Identifier introduction-to-algebra-the-art-of-problem-solving-by-richard-rusczyk Identifier-ark ark:/13960/s2bcrt2bc0p Ocr tesseract 5.3.0-3-g9920 Ocr_autonomous true Ocr_detected_lang en Ocr_detected_lang_conf

  17. Introduction to Algebra 2nd Edition

    Richard Rusczyk founded Art of Problem Solving (AoPS) in 2003 to create interactive educational opportunities for avid math students. Richard is one of the co-authors of the Art of Problem Solving classic textbooks, author of Art of Problem Solving's Introduction to Algebra, Introduction to Geometry, and Precalculus textbooks, co-author of Art of Problem Solving's Intermediate Algebra and ...

  18. Art of Problem Solving

    Getting Started. Getting Started with Competition Math, a textbook meant for true beginners (on-target middle school students, or advanced elementary school students). It is written by AoPS Community Member cargeek9, currently a junior in high school. It covers the basics of algebra, geometry, combinatorics, and number theory, along with sets ...

  19. Art of Problem Solving

    The center of the universe for students who love math.

  20. Introduction to Linear Algebra

    Linear algebra is a foundational branch of mathematics that deals with vector spaces, linear transformations, and systems of linear equations. It provides essential tools and techniques for solving a wide range of problems in mathematics, science, engineering, and computer science. Understanding linear algebra is crucial for grasping advanced ...