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Linear Algebra and Matrix Theory
1st Edition - August 28, 1990
Authors: Jimmie Gilbert, Linda Gilbert
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Intended for a serious first course or a second course, this textbook will carry students beyond eigenvalues and eigenvectors to the classification of bilinear forms, to normal… Read more
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Intended for a serious first course or a second course, this textbook will carry students beyond eigenvalues and eigenvectors to the classification of bilinear forms, to normal matrices, to spectral decompositions, and to the Jordan form. The authors approach their subject in a comprehensive and accessible manner, presenting notation and terminology clearly and concisely, and providing smooth transitions between topics. The examples and exercises are well designed and will aid diligent students in understanding both computational and theoretical aspects. In all, the straightest, smoothest path to the heart of linear algebra.
* Special Features: * Provides complete coverage of central material. * Presents clear and direct explanations. * Includes classroom tested material. * Bridges the gap from lower division to upper division work. * Allows instructors alternatives for introductory or second-level courses.
Real Coordinate Spaces: The Vector Spaces R[super]n. Subspaces of R[super]n. Geometric Interpretations of R[super]2 and R[super]3. Bases and Dimension. Elementary Operations on Vectors: Elementary Operations and Their Inverses. Elementary Operations and Linear Independence. Standard Bases for Subspaces. Matrix Multiplication: Matrices of Transition. Properties of Matrix Multiplication. Invertible Matrices. Column Operationsand Column-Echelon Forms. Row Operations and Row-Echelon Forms. Row and Column Equivalence. Rank and Equivalence. Vector Spaces, Matrices, and Linear Equations: Vector Spaces. Subspaces and Related Concepts. Isomorphisms of Vector Spaces. Standard Bases for Subspaces. Matrices over an Arbitrary Field. Systems of Linear Equations. Linear Transformations: Linear Transformations. Linear Transformations and Matrices. Change of Basis. Composition of Linear Transformations. Determinants: Permutations and Indices. The Definition of a Determinant. Cofactor Expansions. Elementary Operations and Cramer's Rule. Determinants and Matrix Multiplication. Eigenvalues and Eigenvectors: Eigenvalues and Eigenvectors. Eigenspaces and Similarity. Representation by a Diagonal Matrix. Functions of Vectors: Linear Functionals. Real Quadratic Forms. Orthogonal Matrices. Reduction of Real Quadratic Forms. Classification of Real Quadratic Forms. Bilinear Forms. Symmetric Bilinear Forms. Hermitian Forms. Inner Product Spaces: Inner Products. Norms and Distances. Orthonormal Bases. Orthogonal Complements. Isometries. Normal Matrices. Normal Linear Operators. Spectral Decompositions: Projections and Direct Sums. Spectral Decompositions. Minimal Polynomials and Spectral Decompositions. Nilpotent Transformations. The Jordan Canonical Form. Index.