Linear Algebra and Its Applications by David Lay, Steven Lay, Judi McDonald 5th Edition (032198238X) (9780321982384)

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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..

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.



1. Linear Equations in Linear Algebra


Introductory Example: Linear Models in Economics and Engineering


1.1 Systems of Linear Equations


1.2 Row Reduction and Echelon Forms


1.3 Vector Equations


1.4 The Matrix Equation Ax = b


1.5 Solution Sets of Linear Systems


1.6 Applications of Linear Systems


1.7 Linear Independence


1.8 Introduction to Linear Transformations


1.9 The Matrix of a Linear Transformation


1.10 Linear Models in Business, Science, and Engineering


Supplementary Exercises


 


2. Matrix Algebra


Introductory Example: Computer Models in Aircraft Design


2.1 Matrix Operations


2.2 The Inverse of a Matrix


2.3 Characterizations of Invertible Matrices


2.4 Partitioned Matrices


2.5 Matrix Factorizations


2.6 The Leontief Input–Output Model


2.7 Applications to Computer Graphics


2.8 Subspaces of Rn


2.9 Dimension and Rank


Supplementary Exercises


 


3. Determinants


Introductory Example: Random Paths and Distortion


3.1 Introduction to Determinants


3.2 Properties of Determinants


3.3 Cramer’s Rule, Volume, and Linear Transformations


Supplementary Exercises


 


4. Vector Spaces


Introductory Example: Space Flight and Control Systems


4.1 Vector Spaces and Subspaces


4.2 Null Spaces, Column Spaces, and Linear Transformations


4.3 Linearly Independent Sets; Bases


4.4 Coordinate Systems


4.5 The Dimension of a Vector Space


4.6 Rank


4.7 Change of Basis


4.8 Applications to Difference Equations


4.9 Applications to Markov Chains


Supplementary Exercises


 


5. Eigenvalues and Eigenvectors


Introductory Example: Dynamical Systems and Spotted Owls


5.1 Eigenvectors and Eigenvalues


5.2 The Characteristic Equation


5.3 Diagonalization


5.4 Eigenvectors and Linear Transformations


5.5 Complex Eigenvalues


5.6 Discrete Dynamical Systems


5.7 Applications to Differential Equations


5.8 Iterative Estimates for Eigenvalues


Supplementary Exercises


 


6. Orthogonality and Least Squares


Introductory Example: The North American Datum and GPS Navigation


6.1 Inner Product, Length, and Orthogonality


6.2 Orthogonal Sets


6.3 Orthogonal Projections


6.4 The Gram–Schmidt Process


6.5 Least-Squares Problems


6.6 Applications to Linear Models


6.7 Inner Product Spaces


6.8 Applications of Inner Product Spaces


Supplementary Exercises


 


7. Symmetric Matrices and Quadratic Forms


Introductory Example: Multichannel Image Processing


7.1 Diagonalization of Symmetric Matrices


7.2 Quadratic Forms


7.3 Constrained Optimization


7.4 The Singular Value Decomposition


7.5 Applications to Image Processing and Statistics


Supplementary Exercises


 


8. The Geometry of Vector Spaces


Introductory Example: The Platonic Solids


8.1 Affine Combinations


8.2 Affine Independence


8.3 Convex Combinations


8.4 Hyperplanes


8.5 Polytopes


8.6 Curves and Surfaces


 


9. Optimization (Online Only)


Introductory Example: The Berlin Airlift


9.1 Matrix Games


9.2 Linear Programming—Geometric Method


9.3 Linear Programming—Simplex Method


9.4 Duality


 


10. Finite-State Markov Chains (Online Only)


Introductory Example: Googling Markov Chains


10.1 Introduction and Examples


10.2 The Steady-State Vector and Google's PageRank


10.3 Finite-State Markov Chains


10.4 Classification of States and Periodicity


10.5 The Fundamental Matrix


10.6 Markov Chains and Baseball Statistics


 


Appendices


A. Uniqueness of the Reduced Echelon Form


B. Complex Numbers


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