1. LINEAR SYSTEMS AND MATRICES. Linear Systems of Equations. Elementary Operations and Gauss Elimination Method. Homogeneous Linear Systems. Introduction to Matrices and the Matrix of a Linear System. Elementary Row Operations on a Matrix. Proofs of Facts. Chapter Review Questions and Project. 2. ALGEBRA OF MATRICES. Scalar Multiplication and Addition of Matrices. Matrix Multiplication and its Properties. Transpose. Proofs of Facts. Chapter Review Questions and Projects. 3. SUBSPACES. Linear Combination of Vectors. Vector Subspaces. Linear Dependence, Linear Independence and Basis. Proofs of Facts. Chapter Review Questions and Project. 4. RANK. Elementary Operations and Rank. Null Space and Nullity of a Matrix. Elementary Matrices. Proofs of Facts. Chapter Review Questions and Project. 5. INVERSE, RANK FACTORIZATION, AND LU-DECOMPOSITION. Inverse of a Matrix and Its Properties. Further Properties of Inverses. Full-Rank Factorization. LU-Decomposition of a Matrix. Proofs of Facts. Chapter Review Questions and Projects. 6. DETERMINANTS. Determinant. Properties of the Determinant. Cofactors and Inverse of a Matrix. Cramer's Rule. Chapter Review Questions and Projects. 7. EIGENVALUE PROBLEMS. Eigenvalues and Eigenvectors. Characteristic Polynomial. Calculating Eigenvalues and Eigenvectors (Another Approach) and the Cayley-Hamilton. Applications of the Cayley-Hamilton Theorem/ Properties of Eigenvalues, Diagonalizability, and Triangularizability. Proofs and Facts. Chapter Review Questions and Project. 8. INNER PRODUCT SPACES. Gram-Schmidt Orthogonalization Process. Diagonalization of Symmetric Matrices. Application of the Spectral Theorem. Least-Squares Solution. Generalized Inverse and Least-Squares Solution. Proofs and Facts. Chapter Review Questions and Projects. 9. VECTOR SPACES AND LINEAR MAPPINGS. Vector Spaces. Linear Dependence and Linear Independence. Linear Mappings. Some Properties of Linear Mappings:Image and Kernel. Linear Mappings and Matrices. Chapter Review Questions and Project. 10. DETERMINANTS (REVISITED). Permutations. Determinants. Cofactor Expansion. Adjoint of a Matrix. Cramer's Rule. Product Theorem of Determinants. Answers and Hints to Selected Exercises. Drill Solutions Using Matlab. Some Basic Matlab Operations. Index.
About the Author
S. K. Jain is Distinguished Professor of Mathematics at Ohio University.
A.D. Gunawardena is Associate Teaching Professor of Computer Science at Carnegie Mellon University.