Nicholson W. Keith. Linear Algebra with Applications

3rd Edition. — University of Calgary. — Boston: PWS Publishing Company, 1995. — 554 p.This textbook is a basic introduction to the ideas and techniques of linear algebra for first- or second-year students who have a working knowledge of high school algebra. Its aim is to achieve a balance among the computational skills, theory, and application of linear algebra, while keeping the level suitable for beginning students. The contents are aranged to permit enough flexibility to allow the presentation of a traditional introduction to the subject, or to allow a more applied course. Calculus is not prerequisite; places where it is mentioned are clearly marked and may be omitted.Linear algebra has a wide application to the mathematical and natural sciences, to engineering, to computer science, and (increasingly) to management and social sciences. As a rule, students of linear algebra learn the subject by studying examples and solving problems. More than 330 solved examples are included here, many of a computational nature, together with a wide variety of exercises. In addition, a number of section are devoted to applications and to the computational side of the subject. There are optional, but they are included at the end of the relevant chapter (rather than at the end of the book) to encourage students to browse.Systems of Linear Equations
Solutions and Elementary Operations
Gaussian Elimination
Homogeneous Equations
An Application to Network Flow (Optional)
An Application to Electrical Network (Optional)Matrix Algebra
Matrix Addition, Scalar Multiplication, and Transposition
Matrix Multiplication
Matrix Inverses
Elementary Matrices
LU-Factorization (Optional)
An Application to Input-Output Economic Model (Optional)
An Application to Markov Chains (Optional)Determinants
The Laplace Expansion
Determinants and Matrix Inverses
An Application to Polynomial Interpolation (Optional)
Proof of the Laplace Expansion (Optional)Vector Geometry
Vectors and Lines
The Dot Product and Projections
Plane and the Cross Product
An Application to Least Square Approximation (Optional)Vector Spaces
Examples and Basic Properties
Subspaces and Spanning Sets
Linear Independence and Dimension
Existence of Bases
Rank of Matrix
An Application to Polynomials (Optional)
An Application to Differential Equations (Optional)Eigenvalues and Diagonalization
Eigenvalue and Similarity
Diagonalization
Orthogonality in Rⁿ
Orthogonal Diagonalization
Positive Definite Matrices
LP-Factorization (Optional)
Computing Eigenvalues (Optional)
Complex Matrices (Optional)
An Application to Quadratic Forms (Optional)
An Application to Best Approximation and Least Squares (Optional)
An Application to Systems of Differential Equations (Optional)Linear Transformations
Examples and Elementary Properties
Kernel and Image of a Linear Transformation
Isomorphism and Composition
The Matrix of Linear Transformation
Change of Basis
Invariant Subspace and Direct Sums
Block Triangular Form
An Application to Linear Recurrence Relation (Optional)Inner Product Spaces
Inner Products and Norms
Orthogonal Sets of Vectors
Orthogonal Diagonalization
Isometries
An Application to Fourier ApproximationAppendix A. Complex Numbers
Appendix B. Introduction to Linear Programming
Graphical Methods
Simplex Algorithm
Appendix C. Mathematical Induction
Selected Answers
Index