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.
Undergraduate math majors who have some background in linear algebra and would benefit from a strong foundation for more abstract treatments at a higher level. Also serves as text for introductory 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.
- No. of pages:
- © Academic Press 1995
- 28th June 2014
- Academic Press
- eBook ISBN:
University of South Carolina at Spartanburg
University of South Carolina at Spartanburg
@qu:"...the material is well-written, it will be easy to teach from, and the students will find it easy to read and study from." @source:-Jackie Garner, University of Arkansas @qu:"Jimmie and Linda Gilbert...present material at the appropriate level in a quick and concise manner....Some...texts do not develop enough mathematical sophistication to enable the transition to junior- or senior-level linear algebra courses. The Gilbert text does!" @source:--Melvyn Jeter, Illinois WesleyanUniversity @qu:"The notation and terminology are clear and concise and the flow of topics and concepts is smooth." @source:-Ed Dixon, Tennessee Technological University @qu:"I am...quite enthusiastic about the arrangement of topics which the Gilberts have chosen....(The Gilberts Chapter 1 plays as a sort of overture, in which the main elements of vector space theory are laid out in a comfortable space, while due attention is paid to the role of computation. I think this type of start to a course would orient the students properly toward general vector space theory, rather than having them come upon the annoying complication of vector spaces after three or more weeks spent exclusively in computation and row reduction." @source:-Edward Hinson, University of New Hampshire