Linear Algebra and Its Applications – David C. Lay – 4th Edition


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. David 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.

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  • 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: Readjusting the North American Datum
    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: Google and 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

    A. Uniqueness of the Reduced Echelon Form
    B. Complex Numbers
  • Citation
    • Full Title: Linear Algebra and Its Applications
    • Author/s:
    • ISBN-10: 0321385179
    • ISBN-13: 9780321385178
    • Edition: 4th Edition
    • Topic: Math
    • Subtopic: Linear Algebra
    • File Type: eBook
    • Idioma: English

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