1. Matrices and Systems of Equations 1.1 Systems of Linear Equations 1.2 Row Echelon Form 1.3 Matrix Arithmetic 1.4 Matrix Algebra 1.5 Elementary Matrices 1.6 Partitioned Matrices    Matlab Exercises    Chapter Test A    Chapter Test B   2. Determinants 2.1 The Determinant of a Matrix 2.2 Properties of Determinants 2.3 Additional Topics and Applications    Matlab Exercises    Chapter Test A    Chapter Test B   3. Vector Spaces 3.1 Definition and Examples 3.2 Subspaces 3.3 Linear Independence 3.4 Basis and Dimension 3.5 Change of Basis 3.6 Row Space and Column Space    Matlab Exercises    Chapter Test A    Chapter Test B   4. Linear Transformations 4.1 Definition and Examples 4.2 Matrix Representations of Linear Transformations 4.3 Similarity    Matlab Exercises    Chapter Test A    Chapter Test B   5. Orthogonality 5.1 The Scalar Product in Rn 5.2 Orthogonal Subspaces 5.3 Least Squares Problems 5.4 Inner Product Spaces 5.5 Orthonormal Sets 5.6 The Gram—Schmidt Orthogonalization Process 5.7 Orthogonal Polynomials    Matlab Exercises    Chapter Test A    Chapter Test B   6. Eigenvalues 6.1 Eigenvalues and Eigenvectors 6.2 Systems of Linear Differential Equations 6.3 Diagonalization 6.4 Hermitian Matrices 6.5 The Singular Value Decomposition 6.6 Quadratic Forms 6.7 Positive Definite Matrices 6.8 Nonnegative Matrices    Matlab Exercises    Chapter Test A    Chapter Test B   7. Numerical Linear Algebra 7.1 Floating-Point Numbers 7.2 Gaussian Elimination 7.3 Pivoting Strategies 7.4 Matrix Norms and Condition Numbers 7.5 Orthogonal Transformations 7.6 The Eigenvalue Problem 7.7 Least Squares Problems    Matlab Exercises    Chapter Test A    Chapter Test B   Appendix: MATLAB Bibliography A. Linear Algebra and Matrix Theory B. Applied and Numerical Linear Algebra C. Books of Related Interest   Answers to Selected Exercises

Extensive applications of linear algebra concepts to a variety of real world situations. These applications introduce new material and show relevance of the material covered. Students learn how theories and concepts of linear algebra can help solve modern day problems. Interesting and current examples include the application of linear transformations to an airplane, eigenvectors determining the orientation of a space shuttle, and how Google Inc. makes use of linear algebra to rank and order search results. Abundant computer exercises, more extensive than any other linear algebra book on the market, help students to visualise and discover linear algebra and allow them to explore more realistic applications that are too computationally intensive to work out by hand. These exercises also provide students with experience in performing matrix computations. Worked out examples illustrate new concepts, making the material less abstract and helping students quickly build their understanding. Two chapter tests for every chapter help students study for exams and get the practice they need to master the material. A comprehensive MATLAB® appendix gives users all the information they need to use the latest version of MATLAB.  

New applications added to Chapters 1, 5, 6, and 7–in Chapter 1 (Matrices and Systems of Equations), the authors introduce an important application to the field of Management Science. Management decisions often involve making choices between a number of alternatives. The authors assume that the choices are to be made with a fixed goal in mind and should be based on a set of evaluation criteria. These decisions often involve a number of human judgments that may not always be completely consistent. The analytic hierarchy process is a technique for rating the various alternatives based on a chart consisting of weighted criteria and ratings that measure how well each alternative satisfies each of the criteria. Chapter 1 (Matrices and Systems of Equations), shows how to set up such a chart or decision tree for the process. After weights and ratings have been assigned to each entry in the chart, an overall ranking of the alternatives is calculated using simple matrix-vector operations. Chapters 5 (Orthogonality) and 6 (Eigenvalues) revisit the application and discuss how to use advanced matrix techniques to determine appropriate weights and ratings for the decision process. Finally, Chapter 7 (Numerical Linear Algebra) presents a numerical algorithm for computing the weight vectors used in the decision process. New subsection in Chapter 3, Section 2 (Subspaces)–one important example of a subspace occurs when finding all solutions to a homogeneous system of linear equations. This type of subspace is referred to as a nullspace. A new subsection has been added to show how the nullspace is also useful in finding the solution set to a nonhomogeneous linear system. The subsection contains a new theorem and also a new figure that provides a geometric illustration of the theorem. Three related problems have been added to the exercises at the end of Section 2. Chapter 7, Section 1 (Floating-Point Numbers) revised–Two subsections have been revised and modernised. A new subsection on IEEE floating-point representation of numbers and a second subsection on accuracy and stability of numerical algorithms have been added. New examples and additional exercises on these topics were also included. Chapter 7, Section 5 (Orthogonal Transformations) revised–The discussion of Householder transformations has been revised and expanded. A new subsection has been added, which discusses the practicalities of using QR factorisations for solving linear systems. New exercises have also been added to this section. Chapter 7, Section 7 (Least Squares Problems) revised–This revised section deals with numerical methods for solving least squares problems. The section has been revised and a new subsection on using the modified Gram—Schmidt process to solve least squares problems has been added. The subsection contains one new algorithm and some new exercises.