Linear Algebra

1. Systems of Linear Equations.- 1.1 Lines and Linear Equations.- 1.2 Linear Systems and Matrices.- 1.3 Applications of Linear Systems  1.4 Numerical Solutions.- 2. Euclidean Space.- 2.1 Vectors.- 2.2 Span.- 2.3 Linear Independence.- 3. Matrices.- 3.1 Linear Transformations.- 3.2 Matrix Algebra.- 3.3 Inverses.- 3.4 LU Factorization.- 3.5 Markov Chains.- 4. Subspaces.- 4.1 Introduction to Subspaces.- 4.2 Basis and Dimension.- 4.3 Row and Column Spaces.- 4.4 Change of Basis.- 5. Determinants.- 5.1 The Determinant Function.- 5.2 Properties of the Determinant.- 5.3 Applications of the Determinant.- 6. Eigenvalues and Eigenvectors.- 6.1 Eigenvalues and Eigenvectors.- 6.2 Diagonalization.- 6.3 Complex Eigenvalues and Eigenvectors.- 6.4 Systems of Differential Equations .- 6.5 Approximation Methods.- 7. Vector Spaces.- 7.1 Vector Spaces and Subspaces.- 7.2 Span and Linear Independence.- 7.3 Basis and Dimension.- 8. Orthogonality.- 8.1 Dot Products and Orthogonal Sets.- 8.2 Projection and the Gram-Schmidt Process.- 8.3 Diagonalizing Symmetric Matrices and QR Factorization 8.4 The Singular Value Decomposition.- 8.5 Least Squares Regression.- 9. Linear Transformations.- 9.1 Definition and Properties.- 9.2 Isomorphisms.- 9.3 The Matrix of a Linear Transformation.- 9.4 Similarity.- 10. Inner Product Spaces.- 10.1 Inner Products.- 10.2 The Gram-Schmidt Process Revisited.- 10.3 Applications of Inner Products.- 11. Additional Topics and Applications.- 11.1 Quadratic Forms.- 11.2 Positive Definite Matrices.- 11.3 Constrained Optimization.- 11.4 Complex Vector Spaces.- 11.5 Hermitian Matrices.- Glossary.- Answers to Selected Exercises.- Index