Linear Algebra with Applications

1. Linear Systems and Matrices. Geometric View of Linear Systems. Gaussian and Gauss-Jordan Elimination. Row Equivalence and Echelon Matrices. Homogeneous Systems. Matrix Algebra. Properties of Matrix Operations. Row Equivalence and Matrix Multiplication. The LU Factorization (Optional) Partial Pivoting (Optional). 2. Determinants. Definitions and Examples. Evaluation of det (A): A Better Way. Additional Properties of Determinants. Eigenvalues and Eigenvectors. 3. Vector Spaces. R2 and R3: Old Friends. Euclidean n-space and its Subspaces. Subspaces of Rn -continued. Linear Independence and Dependence in Rn. Basis and Dimension. Orthogonality in Rn. Vector Spaces-The General Concept. Subspaces. Linear Independence, Basis, and Dimension. 4. Linear Transformations. Definitions and Examples. The Range and Null Space of a Linear Transformation. The Algebra of Linear Transformations. Geometric Interpretation. Matrices and Linear Transformations. Change of Basis (Optional). The Change-of-Basis Problem (Optional). 5. Similar Matrices and The Eigenvalue Problem. Diagonalization-Eigenvalues and Eigenvectors. Orthogonal Similarity and Symmetric Matrices. Schur's Theorem (Optional). The Power Method (Optional). 6. Linear Programming. The Geometric Point of View. Different Types of Linear Programming Problems. The Simplex Method. Refinements of the Simplex Method. 7. Selected Applications. Graph theory. Least Squares Approximations. Quadratic Forms. Linear Economic. Appendices. Linear Algebra Software. The Matman Program. The Matalg Program. The Matlab Program. Appendices. Answers to Odd Numbered Exercises. Index.