Elementary Linear Algebra

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Bruce Cooperstein– University of California, Santa Cruz

ISBN-10: 0-9885572-0-7
ISBN-13: 978-0-9885572-0-8
942 Pages
©2019 Worldwide Center of Mathematics, LLC

Digital | FREE

 

Features

 

contents

Introduction

This is an e-textbook for a first course in linear algebra. The topics covered include: Linear Systems, The Vector Space R^n, Matrix Algebra, Determinants, Abstract Vector Spaces, Linear Transformations, Eigenvalues and Eigenvectors, and Orthogonality in R^n. It introduces linear transformations in R^n quite early and uses them to motivate the addition and multiplication of matrices. In additional to the usual theory there are several sections devoted to applications such as Markov chain models, age structured population models, Leontief input-output models, error-correcting codes, linear recurrence relations, systems of differential equations, and the characterization of real quadratic curves and real quadratic surfaces. We also obtain the canonical forms for real 2 x 2 and 3 x 3 matrices.

The book is rigorous in its treatment of the theory and all important results are proved. What separates this book from print treatments of linear algebra and other e-textbooks is the use of the digital environment to create a pedagogical product that supports student understanding. Specifically, without limitations of length we can regularly spiral back to important concepts and algorithms. Thus, each section begins with a subsection, “What you need to know” reviewing definitions and contains a short quiz with links to solutions. Also to facilitate student understanding, which depends on mastery of over 100 concepts, nearly every instance of a fundamental term is linked back to its definition. Likewise, in proofs, citation of previous results (lemmas, theorems, corollaries) are linked to their original statements and proofs. Also in each section there is a subsection. How to do it, where we describe the specific algorithms students will need to enact when assigned exercises. Further, in addition to a large selection of exercises, each section contains numerous challenge exercises (problems) which require knowledge of the theorems and the application of mathematical reasoning.


Features

  • Written by a mathematics professor with over 30 years of teaching experience
  • Down-to-Earth exposition, presented as it would be spoken in class
  • Completely rigorous definitions, statements of theorems, and proofs
  • Hyperlinked table of contents, index, and cross-references.
  • PDF format, compatible with all computers, tablets, and mobile devices

Contents

  • 1.1 Linear Equations and Their Solution
  • 1.2 Matrices and Echelon Forms
  • 1.3 How to Use it: Applications of Linear Systems
  • 2.1 Introduction to Vectors: Linear Geometry
  • 2.2 Vectors and the Space Rn
  • 2.3 The Span of a Sequence of Vectors
  • 2.4 Linear independence in Rn
  • 2.5 Subspaces and Bases of Rn
  • 2.6 The Dot Product in Rn
  • 3.1 Introduction to Linear Transformations and Matrix Multiplication
  • 3.2 The Product of a Matrix and a Vector
  • 3.3 Matrix Addition and Multiplication
  • 3.4 Invertible Matrices
  • 3.5 Elementary Matrices
  • 3.6 The LU Factorization
  • 3.7 How to Use It: Applications of Matrix Multiplication
  • 4.1 Introduction to Determinants
  • 4.2 Properties of Determinants
  • 4.3 The Adjoint of a Matrix and Cramer’s Rule
  • 5.1 Introduction to Abstract Vector Spaces
  • 5.2 Span and Independence in Vector Spaces
  • 5.3 Dimension of a finite generated vector space
  • 5.4 Coordinate vectors and change of basis
  • 5.5 Rank and Nullity of a Matrix
  • 5.6 Complex Vector Spaces
  • 5.7 Vector Spaces Over Finite Fields
  • 5.8 How to Use it: Error Correcting Codes
  • 6.1 Introduction to Linear Transformations on Abstract Vector Spaces
  • 6.2 Range and Kernel of a Linear Transformation
  • 6.3 Matrix of a Linear Transformation
  • 7 Eigenvalues and Eigenvectors
  • 7.1 Introduction to Eigenvalues and Eigenvectors
  • 7.2 Diagonalization of Matrices
  • 7.3 Complex Eigenvalues of Real Matrices
  • 7.4 How to Use It: Applications of Eigenvalues and Eigenvectors
  • 8.1 Orthogonal and Orthonormal Sets in Rn
  • 8.2 The Gram-Schmidt Process and QR-Factorization
  • 8.3 Orthogonal Complements and Projections
  • 8.4 Diagonalization of Real Symmetric Matrices
  • 8.5 Quadratic Forms, Conic Sections and Quadratic Surfaces
  • 8.6 How to Use It: Least Squares Approximation