Linear Algebra and Its Applications – David C. Lay – 3rd Edition

Description

is relatively easy for during the early stages of the , when the material is presented in a familiar, 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 ), are not easily understood, and require time to assimilate.

Since they are fundamental to the study of linear , students’ understanding of these concepts is vital to their mastery of the subject. 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.

Table of Contents

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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