Synopses & Reviews
KEY BENEFIT: This trusted reference offers an intellectually honest, thought-provoking, sound introduction to linear algebra. Enables readers to grasp the subject with a challenging, yet visually accessible approach that does not sacrifice mathematical integrity. Adds over 400 new exercises to the problem sets, ranging in difficulty from elementary to more challenging. Adds new historical problems taken from ancient Chinese, Indian, Arabic, and early European sources. Strengthens geometric and conceptual emphasis. A comprehensive, thorough reference for anyone who needs to brush up on their knowledge of linear algebra.
Table of Contents
1. Linear Equations
1.1 Introduction to Linear Systems
1.2 Matrices, Vectors, and Gauss-Jordan Elimination
1.3 On the Solutions of Linear Systems; Matrix Algebra
2. Linear Transformations
2.1 Introduction to Linear Transformations and Their Inverses
2.2 Linear Transformations in Geometry
2.3 Matrix Products
2.4 The Inverse of a Linear Transformation
3. Subspaces of R" and Their Dimensions
3.1 Image and Kernel of a Linear Transformation
3.2 Subspace of R"; Bases and Linear Independence
3.3 The Dimension of a Subspace of R"
3.4 Coordinates
4. Linear Spaces
4.1 Introduction to Linear Spaces
4.2 Linear Transformations and Isomorphisms
4.3 The Matrix of a Linear Transformation
5. Orthogonality and Least Squares
5.1 Orthogonal Projections and Orthonormal Bases
5.2 Gram-Schmidt Process and QR Factorization
5.3 Orthogonal Transformations and Orthogonal Matrices
5.4 Least Squares and Data Fitting
5.5 Inner Product Spaces
6. Determinants
6.1 Introduction to Determinants
6.2 Properties of the Determinant
6.3 Geometrical Interpretations of the Determinant; Cramer's Rule
7. Eigenvalues and Eigenvectors
7.1 Dynamical Systems and Eigenvectors: An Introductory Example
7.2 Finding the Eigenvalues of a Matrix
7.3 Finding the Eigenvectors of a Matrix
7.4 Diagonalization
7.5 Complex Eigenvalues
7.6 Stability
8. Symmetric Matrices and Quadratic Forms
8.1 Symmetric Matrices
8.2 Quadratic Forms
8.3 Singular Values
9. Linear Differential Equations
9.1 An Introduction to Continuous Dynamical Systems
9.2 The Complex Case: Euler's Formula
9.3 Linear Differential Operators and Linear Differential Equations
Appendix A. Vectors
Answers to Odd-numbered Exercises
Subject Index
Name Index