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Ideal as a reference or quick review of the fundamentals of linear algebra, this book offers amatrix-oriented approach--with more emphasis on Euclidean n-space, problem solving, and applications, and less emphasis on abstract vector spaces. It features a variety of applications, boxed statements of important results, and a large number of numbered and unnumbered examples.Matrices, Vectors, and Systems of Linear Equations. Matrices and Linear Transformations. Determinants. Subspaces and Their Properties. Eigenvalues, Eigenvectors, and Diagonalization. Orthogonality. Vector Spaces. Complex Numbers.A professional reference for computer scientists, statisticians, and some engineers.
Table of Contents
1. Matrices, Vectors, and Systems of Linear Equations.
Matrices and Vectors. Linear Combinations, Matrix-Vector Products, and Special Matrices. Systems of Linear Equations. Gaussian Elimination. Applications of Systems of Linear Equations. The Span of a Set Vectors. Linear Dependence and Independence. Chapter 1 Review.
2. Matrices and Linear Transformations.
Matrix Multiplication. Applications of Matrix Multiplication. Invertibility and Elementary Matrices. The Inverse of a Matrix. The LU Decomposition of a Matrix. Linear Transformations and Matrices. Composition and Invertibility of Linear Transformations. Chapter 2 Review.
3. Determinants.
Cofactor Expansion. Properties of Determinants. Chapter 3 Review.
4. Subspaces and Their Properties.
Subspaces. Basis and Dimension. The Dimension of Subspaces Associated with a Matrix. Coordinate Systems. Matrix Representations of Linear Operators. Chapter 4 Review.
5. Eigenvalues, Eigenvectors, and Diagonalization.
Eigenvalues and Eigenvectors. The Characteristic Polynomial. Diagonalization of Matrices. Diagonalization of Linear Operators. Applications of Eigenvalues. Chapter 5 Review.
6. Orthogonality.
The Geometry of Vectors. Orthonormal Vectors. Least-Squares Approximation and Orthogonal Projection Matrices. Orthogonal Matrices and Operators. Symmetric Matrices. Singular Value Decomposition. Rotations of R^3 and Computer Graphics. Chapter 6 Review.
7. Vector Spaces.
Vector Spaces and their Subspaces. Dimension and Isomorphism. Linear Tranformations and Matrix Representations. Inner Product Spaces. Chapter 7 Review.