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Systems of Linear Equations
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Lecture Notes on Linear
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Lecture Notes on Linear
Index
Contents
Systems of Linear Equations
Introduction
Terminologies:
Example:
One of the main problems considered
Some Caution
Gauss-Jordan Elimination Method
Reduced Row Echelon Form
Coefficient and Augmented matrices
Abstraction from Gauss-Jordan Elimination Method
Elementary Row Operations
Matrix Product
Square and Elementary Matrices
Invertible Matrices
Application to the Solution sets of
Linear Systems
Arithmetic of Matrices
Vector Spaces
Vector Spaces and Subspaces
Basic Terminologies
Basis
Dimension
Matrix Multiplication: Another View
Row and Column Spaces
Applications
Linear Transformations
The Basics of Linear Transformations
Algebra of Linear Transformations
Representing Linear Mapping as Matrix
Linear Homomorphisms
Changing Bases and Linear Operators
Changing Bases
Algebra of Linear Operators
Dual Spaces and Linear Functionals
Dual Spaces
Geometric Meanings of Linear Functionals
Linear Functionals on
Double Duals
Transpose of a Linear Transformation
Polynomial Algebras and Principal Ideal Domains
Definitions of Algebra, Module, and Ring
Integral Domains
Principal Ideal Domains
Unique Factorization Domains
Decompositions of Polynomials
Multilinear Forms and Determinants
Multilinear Forms
The Graded Tensor Algebra:
Alternating Forms
Alternating
-forms
Alternating
-forms
Determinants
Determinants of Matrices
Determinant as an alternating
-form
Reduction of Determinant Computation
Tools for Determinant Computation
Exterior Algebra
Elementary Canonical Forms
Introduction
Characteristic and Minimal Polynomials
Direct Sums and Projections
Primary Decomposition Theorem
Diagonalizable and Triangulable Linear Operators
The Rational and Jordan Forms
The Companion and Elementary Jordan Matrices
The Elementary Jordan Matrices
Equivalence of Matrices over P.I.D.
Companion and Rational Canonical Matrices
Vector Spaces As Torsion Modules
Free Modules
Modules over a Principal Ideal Domain
Noetherian Modules
The decomposition of finitely generated
-module
Equivalence of Matrices
The Calculation of Invariant Factors
Each Matrix Is Equivalent to an Extended Diagonal Matrix
The Uniqueness of Extended Diagonal Matrix
Applications to Linear Operators
Primary Decomposition Theorem revised
Appendix:
Index
Felix Hsu 2007-02-27