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### Summary

### Summary

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts are more accessible. Students conceptual understanding is reinforced through True/False questions, practice problems, and the use of technology. David Lay changed the study of linear algebra with the execution of this philosophy, and continues his quest to improve the way linear algebra is taught with the new Updated Second Edition. With this update he takes the instruction of linear algebra to a new level through inc

### Table of Contents

(Each chapter begins with an Introductory Example and ends with Supplementary Exercises.) |

1 Linear Equations In Linear Algebra |

Introductory Example: Linear Models in Economics and Engineering |

Systems of Linear Equations |

Row Reduction and Echelon Forms |

Vector Equations |

The Matrix Equation Ax = b |

Solution Sets of Linear Systems |

Linear Independence |

Introduction to Linear Transformations |

The Matrix of a Linear Transformation |

Linear Models in Business, Science, and Engineering |

Supplementary Exercises |

2 Matrix Algebra |

Introductory Example: Computer Graphics in Automotive Design |

Matrix Operations |

The Inverse of a Matrix |

Characterizations of Invertible Matrices |

Partitioned Matrices |

Matrix Factorizations |

Iterative Solutions of Linear Systems |

The Leontief Input-Output Model |

Applications to Computer Graphics |

Subspaces of Rn |

Supplementary Exercises |

3 Determinants |

Introductory Example: Determinants in Analytic Geometry |

Introduction to Determinants |

Properties of Determinants |

Cramer?s Rule, Volume, and Linear Transformations |

Supplementary Exercises |

4 Vector Spaces |

Introductory Example: Space Flight and Control Systems |

Vector Spaces and Subspaces |

Null Spaces, Column Spaces, and Linear Transformations |

Linearly Independent Sets; Bases |

Coordinate Systems |

The Dimension of a Vector Space |

Rank |

Change of Basis |

Applications to Difference Equations |

Applications to Markov Chains |

Supplementary Exercises |

5 Eigenvalues and Eigenvectors |

Introductory Example: Dynamical Systems and Spotted Owls |

Eigenvectors and Eigenvalues |

The Characteristic Equation |

Diagonalization |

Eigenvectors and Linear Transformations |

Complex Eigenvalues |

Discrete Dynamical Systems |

Applications to Differential Equations |

Iterative Estimates for Eigenvalues |

Supplementary Exercises |

6 Orthogonality and Least-Squares |

Introductory Example: Readjusting the North American Datum |

Inner Product, Length, and Orthogonality |

Orthogonal Sets |

Orthogonal Projections |

The Gram-Schmidt Process |

Least-Squares Problems |

Applications to Linear Models |

Inner Product Spaces |

Applications of Inner Product Spaces |

Supplementary Exercises |

7 Symmetric Matrices and Quadratic Forms |

Introductory Example: Multichannel Image Processing |

Diagonalization of Symmetric Matrices |

Quadratic Forms |

Constrained Optimization |

The Singular Value Decomposition |

Applications to Image Processing and Statistics |

Supplementary Exercises |

Appendices |

Uniqueness of the Reduced Echelon Form |

Complex Numbers |

Glossary |

Answers to Odd-Numbered Exercises |

Index |