Linear Algebra and Its Applications / Edition 4by David C. Lay
Pub. Date: 02/03/2011
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With traditional linear algebra texts, the course is relatively easy for students during the early stages as material is presented in a familiar, concrete setting. However, when abstract concepts are introduced, students often hit a 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. These concepts are fundamental to the study of linear algebra, so students' understanding of them is vital to mastering the subject. This text makes these concepts more accessible by introducing them early in a familiar, concrete Rn setting, developing them gradually, and returning to them throughout the text so that when they are discussed in the abstract, students are readily able to understand.
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Table of Contents
(Each chapter begins with an Introductory Example and ends with Supplementary Exercises.)
1. Linear Equations In Linear Algebra.
Introductory Example: Linear Models in Economics and Engineering.
Systems of Linear Equations.
Row Reduction and Echelon Forms.
The Matrix Equation Ax = b.
Solution Sets of Linear Systems.
Introduction to Linear Transformations.
The Matrix of a Linear Transformation.
Linear Models in Business, Science, and Engineering.
Supplementary Exercises .
2. Matrix Algebra.
Introductory Example: Computer Graphics in Automotive Design.
The Inverse of a Matrix.
Characterizations of Invertible Matrices.
Iterative Solutions of Linear Systems.
The Leontief Input-Output Model.
Applications to Computer Graphics.
Subspaces of Rn.
Supplementary Exercises .
Introductory Example: Determinants in Analytic Geometry.
Introduction to Determinants.
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.
The Dimension of a Vector Space.
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.
Eigenvectors and Linear Transformations.
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.
The Gram-Schmidt Process.
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.
The Singular Value Decomposition.
Applications to Image Processing and Statistics.
Uniqueness of the Reduced Echelon Form.
Answers to Odd-Numbered Exercises.
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