Elementary Linear Algebra

Bruce Cooperstein

 

 

 

Elementary Linear Algebra

Elementary Linear Algebra
Bruce Cooperstein – University of California, Santa Cruz

ISBN-10: 0-9885572-0-7
ISBN-13: 978-0-9885572-0-8
942 Pages
©2015 Worldwide Center of Mathematics

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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.

 

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Contents

1 Linear Equations 1
1.1 Linear Equations and Their Solution
1.2 Matrices and Echelon Forms
1.3 How to Use it: Applications of Linear Systems

2 The Vector Space Rn
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 Matrix Algebra
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 Determinants
4.1 Introduction to Determinants
4.2 Properties of Determinants
4.3 The Adjoint of a Matrix and Cramer’s Rule

5 Abstract Vector Spaces
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 Linear Transformations

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 Orthogonality in Rn

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

 

Features

 

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Supplements

 

Elementary Linear Algebra Recipe Book — This book is meant as a complement to whatever traditional linear algebra text- book has been chosen for your course. The entire book is devoted to the methods, procedures, and algorithms for solving the computational/numerical exercises that are generally asked of students in an elementary linear algebra course. More specifically, you will find within these pages over one hundred such procedures and methods, each described in simple language and illustrated with multiple examples, over 200 in all.

 

Author

 

Bruce Cooperstein
Bruce Cooperstein received his Ph.D. in mathematics from the University of Michigan in 1975. He has been on the faculty of the University of California, Santa Cruz continuously since 1975, obtaining the rank of full professor in 1989. He has won two prestigious awards, a W.K.Kellogg National Fellowship (1982-85) and a Pew National Scholarship for Carnegie Scholars (1999-2000). Bruce’s research areas include finite groups, groups of Lie Type, Lie geometries, incidence and Galois geometry. He is author of one of the first on-line course portfolios, Learning to Think Mathematically. and is the author of over 50 papers that have appeared in referred journals and proceedings of conferences. Bruce was a visiting Fellow of the Carnegie Foundation in Spring, 2007, and has also been involved in mathematics teacher professional development and mathematics education for over two decades.