Linear Algebra and Its Applications

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2003-01-01
  • Publisher: Addison Wesley
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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 so that when discussed in the abstract, these concepts are more accessible.

Table of Contents

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
Supplementary Exercises
Matrix Algebra
Introductory Example: Computer Models 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
Dimension and Rank
Supplementary Exercises
Introductory Example: Determinants in Analytic Geometry
Introduction to Determinants
Properties of Determinants
Cramer's Rule, Volume, and Linear Transformations
Supplementary Exercises
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 a Vector Space
Change of Basis
Applications to Difference Equations
Applications to Markov Chains
Supplementary Exercises
Eigenvalues and Eigenvectors
Introductory Example: Dynamical Systems and Spotted Owls
Eigenvectors and Eigenvalues
The Characteristic Equation
Eigenvectors and Linear Transformations
Complex Eigenvalues
Discrete Dynamical Systems
Applications to Differential Equations
Iterative Estimates for Eigenvalues
Supplementary Exercises
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
Supplementary Exercises
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
Supplementary Exercises (ONLINE ONLY)
The Geometry of Vector Spaces
Introductory Example: The Platonic Solids
Affine Combinations
Affine Independence
Convex Combinations
Curves and Surfaces
Supplementary Exercises (ONLINE ONLY)
Introductory Example: The Berlin Airlift
Matrix Games
Linear Programming - Geometric Method
Linear Programming - Simplex Method
Supplementary Exercises
Uniqueness of the Reduced Echelon Form
Complex Numbers
Answers to Odd-Numbered Exercises
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