Linear Algebra
Linear algebra is a branch of mathematics concerned with the study of vectors, vector spaces, linear transformations, and systems of linear equations. Vector spaces are very important in modern mathematics. Linear algebra is widely used in abstract algebra and functional analysis. It has extensive applications in natural and social sciences, for both linear systems and linear models of nonlinear systems.
It is part of the study of Abstract algebra.
Contents |
Contents
This book is part of a series on Algebra:
Algebra I (Simple)
Algebra I
Algebra
Intermediate Algebra
Linear Algebra
Abstract Algebra
Algebra 2
Determinants
- Fields
- Systems of Linear Equations
- Matrices and Determinants
- Cofactors and Minors
- Cramer's Rule
- Lapace's Theorem
- Linear Dependance
- Gaussian elimination
Vector Spaces
- Vector Spaces
- Linear Dependance
- Bases and dimensions
- Subspaces
- Direct Sum
- Quotient Space
- Span of a set
- Hyperplanes
- Tensors
Systems of Linear Equations
Linear Transformations
- Linear transformations
- Matrices
- Invertible matrices
- The algebra of linear transformations
- The transpose of a linear transformation
- Linear functionals
- Null spaces
- Invariant Subspaces
- Eigenvectors and Eigenvalues
Coordinate Transformations
- Basis Transformations
- Vector Transformations
- Linear Form Transformations
- Matrix Transformations
- Tensor Transformations
Canonical Form
- Nilpotent Operators
- General Canonical Form
- Elementary Divisors
- Jordan Canonical Form
- Operator Functions
Bilinear Form
Quadratic Form
- Quadratic Forms
- Quadratic Forms and Canonical Forms
- Jacobi's Mehtod
- Multilinear Forms
- Basic Theorem
- Extremal Properties
- Quadratic Surfaces
Euclidean Space
Unitary Space
Algebras
- Algebras
- Abstract Algebras
- Schur's Lemma
- Types of Algebras
- Left Regular Representation
- Simple Algebras
- Semisimple Algebras
General Information and MoS
This book is meant for students who wish to study linear algebra from scratch. The approach will not be entirely informal. Every result in the book is intended to be either proved or justified by some mathematical procedure. Links to tedious proofs can be made to The Book of Mathematical Proofs/Algebra after the proof is written there. Terms which are being defined for the first time should be italicized.
Exercises
Learning to think is extremely important in mathematics. Therefore in this book exercises form an important component and by no means should be ignored. Many important concepts of linear algebra are developed via the exercises in the book. It is necessary that before proceeding to the next chapter, the student does the exercises. Links to hints and solutions to many of the exercises are provided but they should be only used in cases of difficulty.
Old pages to be implemented
- Null Spaces
(May 25, 2007) - Column and Row Spaces (May 25, 2007)
- Systems of Linear Equations
(Oct 9, 2006) - Row Reduction and Echelon Forms (Apr 13, 2007)
- The Matrix Equation Ax=b
- Inner Product Spaces
(May 11, 2007) - Matrices and Vectors
- Eigenvalues and Eigenvectors
- Zero Matrices and Zero Vectors (Apr 13, 2007)
- Characteristic Equation
- Matrix Operations (Jun 27, 2007)
- Cramer's Rule
(Oct 20, 2006) - Change of Basis (Jun 27, 2007)
- Introduction to Determinants (May 11, 2007)
- The Inverse of a Matrix (May 11, 2007)
- Partitioned Matrices (Apr 13, 2007)
- Vector Spaces And Subspaces (Need attention as this is a page from the old book)
- Inner Product, Length, and Orthogonality (It is a stub)
- Orthogonal Sets
- Augmented Matrices (Oct 17, 2006)
- Linear Transformations (May 11, 2007)
- Basis Vectors (Oct 20, 2006)
- Linear Transformations (should be looked through, and maybe rewritten as it is taken from the "old" book) (Oct 20, 2006)
- Glossary
- Matrices
- Vectors
- Linear transformations
- Eigenvalues and eigenvectors
External links
- A course in linear algebra - It contains a free set of video lectures given at the Massachusetts Institute of Technology.
- A free book on linear algebra - by Jim Hefferon (St. Michael's College)
- A First Course in Linear Algebra - a free textbook by Rob Beezer (University of Puget Sound) GNU
- Lecture Notes on Linear Algebra
- A toolkit for linear algebra students - This page is designed to help a linear algebra student learn and practice a basic linear algebra procedure, such as Gauss-Jordan reduction, calculating the determinant, or checking for linear independence. By Przemyslaw Bogacki. Department of Mathematics and Statistics, Old Dominion university.
