A Concise Introduction to Linear Algebra

Building on the author's previous edition on the subject (Introduction toLinear Algebra, Jones & Bartlett, 1996), this book offers a refreshingly concise text suitable for a standard course in linear algebra, presenting a carefully selected array of essential topics that can be thoroughly covered in a single semester. Although the exposition generally falls in line with the material recommended by the Linear Algebra Curriculum Study Group, it notably deviates in providing an early emphasis on the geometric foundations of linear algebra. This gives students a more intuitive understanding of the subject and enables an easier grasp of more abstract concepts covered later in the course.The focus throughout is rooted in the mathematical fundamentals, but the text also investigates a number of interesting applications, including a section on computer graphics, a chapter on numerical methods, and many exercises and examples using MATLAB. Meanwhile, many visuals and problems (a complete solutions manual is available to instructors) are included to enhance and reinforce understanding throughout the book.Brief yet precise and rigorous, this work is an ideal choice for a one-semester course in linear algebra targeted primarily at math or physics majors. It is a valuable tool for any professor who teaches the subject. Cover -- Half title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Chapter 1 Vectors and linear systems -- 1.1 Vectors and linear combinations -- 1.2 Length, angle and dot products -- 1.3 Matrices -- Chapter 2 Solving linear systems -- 2.1 Vectors and linear equations -- 2.2 Matrix operations -- 2.3 Inverse matrices -- 2.4 LU decomposition -- 2.5 Transpose and permutation -- Chapter 3 Vector spaces -- 3.1 Spaces of vectors -- 3.2 Nullspace, row space and column space -- 3.3 Solutions of Ax = b -- 3.4 Rank of matrices -- 3.5 Bases and dimensions of general vector spaces -- Chapter 4 Orthogonality -- 4.1 Orthogonality of the four subspaces -- 4.2 Projections -- 4.3 Least squares approximations -- 4.4 Orthonormal bases and Gram-Schmidt -- Chapter 5 Determinants -- 5.1 Introduction to determinants -- 5.2 Properties of determinants -- Chapter 6 Eigenvalues and eigenvectors -- 6.1 Introduction to eigenvectors and eigenvalues -- 6.2 Diagonalizability -- 6.3 Applications to differential equations -- 6.4 Symmetric matrices and quadratic forms -- 6.5 Positive definite matrices -- Chapter 7 Singular value decomposition -- 7.1 Singular value decomposition -- 7.2 Principal component analysis -- Chapter 8 Linear transformations -- 8.1 Linear transformation and matrix representation -- 8.2 Range and null spaces of linear transformation -- 8.3 Invariant subspaces -- 8.4 Decomposition of vector spaces -- 8.5 Jordan normal form -- 8.6 Computation of Jordan normal form -- Chapter 9 Linear programming -- 9.1 Extreme points -- 9.2 Simplex method -- 9.3 Simplex tableau -- Index __Concise Introduction to Linear Algebra__deals with the subject of linear algebra, covering vectors and linear systems, vector spaces, orthogonality, determinants, eigenvalues and eigenvectors, singular value decomposition. It adopts an efficient approach to lead students from vectors, matrices quickly into more advanced topics including, LU decomposition, orthogonal decomposition, Least squares solutions, Gram-Schmidt process, eigenvalues and eigenvectors, diagonalizability, spectral decomposition, positive definite matrix, quadratic forms, singular value decompositions and principal component analysis. This book is designed for onesemester teaching to undergraduate students. Concise Introduction to Linear Algebra deals with the subject of linear algebra, covering vectors and linear systems, vector spaces, orthogonality, determinants, eigenvalues and eigenvectors, singular value decomposition. It adopts an efficient approach to lead students from vectors, matrices quickly into more advanced topics including, LU decomposition, orthogonal decomposition, Least squares solutions, Gram-Schmidt process, eigenvalues and eigenvectors, diagonalizability, spectral decomposition, positive definite matrix, quadratic forms, singular value decompositions and principal component analysis. This book is designed for onesemester teaching to undergraduate students.