Elementary Linear Algebra (Hardcover, 10 Revised edition)

CHAPTER 1 SYSTEMS OF LINEAR EQUATIONS AND MATRICES. 1.1. Introduction to Systems of Linear Equations. 1.2. Gaussian Elimination. 1.3. Matrices and Matrix Operations. 1.4. Inverses; Algebraic Properties of Matrices. 1.5. Elementary Matrices and a Method for Finding A-1. 1.6. More on Linear Systems and Invertible Matrices . 1.7. Diagonal, Triangular, and Symmetric Matrices. 1.8. Applications of Linear Systems (Traffic flow; Electrical Networks; Balancing Chemical Equations, Polynomial Interpolation. 1.9. Leontief Input-Output Models. Chapter Summary. CHAPTER 2 DETERMINANTS. 2.1. Determinants by Cofactor Expansion. 2.2. Evaluating Determinants by Row Reduction. 2.3. Properties of Determinants; Adjoint; Cramer's Rule. Chapter Summary. CHAPTER 3 EUCLIDEAN VECTOR SPACES. 3.1. Vectors in 2-space, 3-space, and n-space. 3.2. Norm, Dot Product, and Distance in Rn. 3.3. Orthogonality. 3.4. The Geometry of Linear Systems. 3.5. Cross Product. Chapter Summary. CHAPTER 4 GENERAL VECTOR SPACES. 4.1. Real Vector Spaces. 4.2. Subspaces. 4.3. Linear Independence. 4.4. Coordinates and Basis. 4.5. Dimension. 4.6. Change of Basis. 4.7. Row Space, Column Space, and Null Space. 4.8. Rank, Nullity, and the Fundamental Matrix Spaces. 4.9 Linear Transformations from Rn to Rm. 4.10. Properties of Matrix Transformations from Rn to Rm. 4.11. Geometry of Matrix Operators. 4.12. Dynamical Systems and Markov Chains. Chapter Summary. CHAPTER 5 EIGNVALUES AND EIGENVECTORS. 5.1. Eigenvalues and Eigenvectors. 5.2. Diagonalization. 5.3. Complex Vector Spaces. 5.4. Application to Differential Equations. Chapter Summary. CHAPTER 6 INNER PRODUCT SPACES. 6.1. Inner Products. 6.2. Angle and Orthogonality in Inner Product Spaces. 6.3. Orthonormal Bases; Gram-Schmidt Process; QR - Decomposition. 6.4. Best Approximation; Least Squares. 6.5. Least Squares Fitting to Data. 6.6. Fourier Series. Chapter Summary. CHAPTER 7 DIAGONALIZATION AND QUADRATIC FORMS. 7.1. Orthogonal Matrices. 7.2. Orthogonal Diagonalization. 7.3. Quadratic Forms. 7.4. Application of Quadratic Forms to Optimization. 7.5. Hermitian, Unitary, and Normal Matrices. Chapter Summary. CHAPTER 8 LINEAR TRANSFORMATIONS. 8.1. General Linear Transformations. 8.2. Isomorphism. 8.3. Composition and Inverse Transformations. 8.4. Matrices of General Linear Transformations. 8.5. Similarity. Chapter Summary. CHAPTER 9 NUMERICAL METHODS. 9.1. Matrix Factorization and LU-Decompositions. 9.2. The Power Method. 9.3. Application to Internet Search Engines. 9.4. Comparison of Procedures for Solving Linear Systems. 9.5. Singular-Value Decomposition. 9.6. Application of Singular Value Decomposition to Data Compression. Chapter Summary.