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Linear algebra is relatively easy for students when the material is presented in a familiar, concrete setting, but it becomes much more difficult when abstract concepts are introduced. Certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood but are fundamental to the study of linear algebra. The author introduces these concepts early in a familiar, concrete R^{n} setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts more accessible and student understanding is reinforced through True-or-False questions, practice problems, and the use of technologies, such as MATLAB, for solving large matrix exercises.
FEATURES:
Author Bio
Lay, David C. : University of Maryland Baltimore
Chapter 1: Linear Equations in Linear Algebra
Systems of Linear Equations
Row Reduction and Echelon Forms
Vector Equations
The Matrix Equation Ax = b
Solution Sets of Linear Systems
Linear Independence
Introduction to Linear Transformations
The Matrix of a Linear Transformation
Linear Models in Business, Science, and Engineering
Chapter 2: Matrix Algebra
Matrix Operations
The Inverse of a Matrix
Characterizations of Invertible Matrices
Partitioned Matrices
Matrix Factorizations
Iterative Solutions of Linear Systems
The Leontief InputOutput Model
Applications to Computer Graphics
Subspaces of R^{n}
Chapter 3: Determinants
Introduction to Determinants
Properties of Determinants
Cramer's Rule, Volume, and Linear Transformations
Chapter 4: Vector Spaces
Vector Spaces and Subspaces
Null Spaces, Column Spaces, and Linear Transformations
Linearly Independent Sets; Bases
Coordinate Systems
The Dimension of a Vector Space
Rank
Change of Basis
Applications to Difference Equations
Applications to Markov Chains
Chapter 5: Eigenvalues and Eigenvectors
Eigenvectors and Eigenvalues
The Characteristic Equation
Diagonalization
Eigenvectors and Linear Transformations
Complex Eigenvalues
Discrete Dynamical Systems
Applications to Differential Equations
Iterative Estimates for Eigenvalues
Chapter 6: Orthogonality and Least-Squares
Inner Product, Length, and Orthogonality
Orthogonal Sets
Orthogonal Projections
The GramSchmidt Process
Least-Squares Problems
Applications to Linear Models
Inner Product Spaces
Applications of Inner Product Spaces
Chapter 7: Symmetric Matrices and Quadratic Forms
Diagonalization of Symmetric Matrices
Quadratic Forms
Constrained Optimization
The Singular Value Decomposition
Applications to Image Processing and Statistics
Appendices
Uniqueness of the Reduced Echelon Form
Complex Numbers
Linear algebra is relatively easy for students when the material is presented in a familiar, concrete setting, but it becomes much more difficult when abstract concepts are introduced. Certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations) are not easily understood but are fundamental to the study of linear algebra. The author introduces these concepts early in a familiar, concrete R^{n} setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts more accessible and student understanding is reinforced through True-or-False questions, practice problems, and the use of technologies, such as MATLAB, for solving large matrix exercises.
FEATURES:
Author Bio
Lay, David C. : University of Maryland Baltimore
Table of Contents
Chapter 1: Linear Equations in Linear Algebra
Systems of Linear Equations
Row Reduction and Echelon Forms
Vector Equations
The Matrix Equation Ax = b
Solution Sets of Linear Systems
Linear Independence
Introduction to Linear Transformations
The Matrix of a Linear Transformation
Linear Models in Business, Science, and Engineering
Chapter 2: Matrix Algebra
Matrix Operations
The Inverse of a Matrix
Characterizations of Invertible Matrices
Partitioned Matrices
Matrix Factorizations
Iterative Solutions of Linear Systems
The Leontief InputOutput Model
Applications to Computer Graphics
Subspaces of R^{n}
Chapter 3: Determinants
Introduction to Determinants
Properties of Determinants
Cramer's Rule, Volume, and Linear Transformations
Chapter 4: Vector Spaces
Vector Spaces and Subspaces
Null Spaces, Column Spaces, and Linear Transformations
Linearly Independent Sets; Bases
Coordinate Systems
The Dimension of a Vector Space
Rank
Change of Basis
Applications to Difference Equations
Applications to Markov Chains
Chapter 5: Eigenvalues and Eigenvectors
Eigenvectors and Eigenvalues
The Characteristic Equation
Diagonalization
Eigenvectors and Linear Transformations
Complex Eigenvalues
Discrete Dynamical Systems
Applications to Differential Equations
Iterative Estimates for Eigenvalues
Chapter 6: Orthogonality and Least-Squares
Inner Product, Length, and Orthogonality
Orthogonal Sets
Orthogonal Projections
The GramSchmidt Process
Least-Squares Problems
Applications to Linear Models
Inner Product Spaces
Applications of Inner Product Spaces
Chapter 7: Symmetric Matrices and Quadratic Forms
Diagonalization of Symmetric Matrices
Quadratic Forms
Constrained Optimization
The Singular Value Decomposition
Applications to Image Processing and Statistics
Appendices
Uniqueness of the Reduced Echelon Form
Complex Numbers