Abstract
1. Linear systems: Introduction to Linear Systems. Gauss Elimination. Numerical Solutions. 2. Vectors: Vector Operations. Dot Product. Span. Linear Independence. The Product [Ax]. Cross Product. Lines, Planes, and Hyperplanes. 3. Matrices: Matrix Operations. Matrix Inverse. Elementary and Invertible Matrices. LU Factorization. 4. Vector spaces: Subspaces of [R to the n Power]. Vector Spaces. Linear Independence Bases. Dimension. Coordinate Vectors and Change of Basis. Rank and Nullity. Applications to Coding Theory. 5. Linear transformations: Matrix Transformations. Linear Transformations. Kernel and Range. The Matrix of a Linear Transformation. The Algebra of Linear Transformations. 6. Determinants: Determinants Cofactor Expansion. Properties of Determinants. The Adjoint Cramer's Rule. Determinants with Permutations. 7. Eigenvalues and eigenvectors: Eigenvalues and Eigenvectors. Diagonalization. Approximations of Eigenvalues and Eigenvectors. Applications to Dynamical Systems. Applications to Markov Chains. 8. Dot and inner products: Orthogonal Sets and Matrices. Orthogonal Projections Gram-Schmidt Process. The QR Factorization. Least Squares. Orthogonalization of Symmetric Matrices. Quadratic Forms and Conic Sections. The Singular Value Decomposition (SVD). Inner Products. Appendices: Answers to selected exercises Each chapter concludes with Applications, Mini-Projects, and Computer Exercises.
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