Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts are more accessible.
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Höhe: 239 mm
Breite: 210 mm
Dicke: 23 mm
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978-0-201-70970-4 (9780201709704)
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Schweitzer Klassifikation
David C. Lay holds a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. Lay has been an educator and research mathematician since 1966, mostly at the University of Maryland, College Park. He has also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He has over 30 research articles published in functional analysis and linear algebra.
As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group, Lay has been a leader in the current movement to modernize the linear algebra curriculum. Lay is also co-author of several mathematics texts, including Introduction to Functional Analysis, with Angus E. Taylor, Calculus and Its Applications, with L.J. Goldstein and D.I. Schneider, and Linear Algebra Gems-Assets for Undergraduate Mathematics, with D. Carlson, C.R. Johnson, and A.D. Porter.
A top-notch educator, Professor Lay has received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland. In 1994, he was given one of the Mathematical Association of America's Awards for Distinguished College or Unviersity Teaching of Mathematics. He has been elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. Lay is a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics. Since 1992, he has served several terms on the national board of the Association of Christians in the Mathematical Sciences.
(Supplementary Exercises are featured at the end of each chapter.)
1. Linear Equations in Linear Algebra.
Introductory Example: Linear Models in Economics and Engineering.
Systems of Linear Equations.
Row Reduction and Echelon Forms.
Vector Equations.
The Matrix Equation Ax = b.
Solution Sets of Linear Systems.
Applications of Linear Systems.
Linear Independence.
Introduction to Linear Transformations.
The Matrix of a Linear Transformation.
Linear Models in Business, Science, and Engineering.
2. Matrix Algebra.
Introductory Example: Computer Graphics in Aircraft Design.
Matrix Operations.
The Inverse of a Matrix.
Characterizations of Invertible Matrices.
Partitioned Matrices.
Matrix Factorizations.
The Leontief Input-Output Model.
Applications to Computer Graphics.
Subspaces of Rn.
Dimensions and Rank.
3. Determinants.
Introductory Example: Determinants in Analytic Geometry.
Introduction to Determinainants.
Properties of Determinants.
Cramer's Rule, Volume, and Linear Transformations.
4. Vector Spaces.
Introductory Example: Space Flight and Control Systems.
Vector Spaces and Subspaces.
Null Spaces, Column Spaces, and Linear Transformations.
Linearly Independent Sets; Bases.
Coordinate Systems.
The Dimension of Vector Space
Rank.
Change of Basis.
Applications to Difference Equations.
Applications to Markov Chains.
5. Eigenvalues and Eigenvectors.
Introductory Example: Dynamical Systems and Spotted Owls.
Eigenvectors and Eigenvalues.
The Characteristic Equation.
Diagonalization.
Eigenvectors and Linear Transformations.
Complex Eigenvalues.
Discrete Dynamical Systems.
Applications to Differential Equations.
Iterative Estimates for Eigenvalues.
6. Orthogonality and Least-Squares.
Introductory Example: Readjusting the North American Datum.
Inner Product, Length, and Orthogonality.
Orthogonal Sets.
Orthogonal Projections.
The Gram-Schmidt Process.
Least-Squares Problems.
Applications to Linear Models.
Inner Product Spaces.
Applications of Inner Product Spaces.
7. Symmetric Matrices and Quadratic Forms.
Introductory Example: Multichannel Image Processing.
Diagonalization of Symmetric Matrices.
Quadratic Forms.
Constrained Optimization.
The Singular Value Decomposition.
Applications to Image Processing and Statistics.
Appendices.
A. Uniqueness of the Reduced Echelon Form.B. Complex NumbersGlossary.Answers.Index.
