Synopses & Reviews
This book presents an introduction to linear algebra and to some of its significant applications. It covers the essentials of linear algebra (including Eigenvalues and Eigenvectors) and shows how the computer is used for applications. Emphasizing the computational and geometrical aspects of the subject, this popular book covers the following topics comprehensively but not exhaustively: linear equations and matrices and their applications; determinants; vectors and linear transformations; real vector spaces; eigenvalues, eigenvectors, and diagonalization; linear programming; and MATLAB for linear algebra. Its useful and comprehensive appendices make this an excellent desk reference for anyone involved in mathematics and computer applications.
Table of Contents
1. Linear Equations and Matrices.
Linear Systems. Matrices. Dot Product and Matrix Multiplication. Properties of Matrix Operations. Matrix Transformations. Solutions of Linear Systems of Equations. The Inverse of a Matrix. LU-Factorization (Optional).
2. Applications of Linear Equations and Matrices (Optional).
An Introduction to Coding. Computer Graphics. Graph Theory. Electrical Circuits. Markov Chains. Linear Economic Models. Introduction to Wavelets.
3. Determinants.
Definition and Properties. Cofactor Expansion and Applications. Determinants from a Computational Point of View.
4. Vectors in Rn .
Vectors in the Plane. n-Vectors. Linear Transformations.
5. Applications of Vectors in R2 and R3 (Optional).
Cross Products in R3. Lines and Planes.
6. Real Vector Spaces.
Real Vector Spaces. Subspaces. Linear Independence. Basis and Dimension. Homogeneous Systems. The Rank of a Matrix and Applications. Coordinates and Change of Basis. Orthonormal Bases in Rn. Orthogonal Complements.
7. Applications of Real Vector Spaces (Optional).
QR-Factorization. Least Square Lines. More on Coding.
8. Eigenvalues, Eigenvectors, and Diagonalization.
Eigenvalues and Eigenvectors. Diagonalization and Similar Matrices. Diagonalization of Symmetric Matrices.
9. Applications of Eigenvalues and Eigenvectors (Optional).
The Fibonacci Sequence. Differential Equations (Calculus Required). Dynamical Systems (Calculus Required). Quadratic Forms. Conic Sections. Quadric Surfaces.
10. Linear Transformations and Matrices.
Definition and Examples. The Kernel and Range of a Linear Transformation. The Matrix of a Linear Transformation. Introduction to Fractals (Optional).
Cumulative Review of Introductory Linear Algebra.
11. Linear Programming (Optional).
The Linear Programming Problem; Geometric Solution. The Simplex Method. Duality. The Theory of Games.
12. MATLAB for Linear Algebra.
Input and Output in MATLAB. Matrix Operations in MATLAB. Matrix Powers and Some Special Matrices. Elementary Row Operations in MATLAB. Matrix Inverses in MATLAB. Vectors in MATLAB. Applications of Linear Combinations in MATLAB. Linear Transformations in MATLAB. MATLAB Command Summary.
Appendix A: Complex Numbers.
Complex Numbers. Complex Numbers in Linear Algebra.
Appendix B: Further Directions.
Inner Product Spaces (Calculus Required). Composite and Invertible Linear Transformations.
Answers to Odd-Numbered Exercises and Chapter Tests. Index.