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 $n \times n$ 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 ${\cal L}(F^{n}, F) = (F^{n})^{*}$
  • 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
    • $M^{r}(V) = V^{*} \otimes \cdots \otimes V^{*}$
    • The Graded Tensor Algebra: $\bigoplus _{r=0}^{\infty } M^{r}(V)$
  • Alternating Forms
    • Alternating $r$-forms
    • Alternating $n$-forms
  • Determinants
    • Determinants of Matrices
    • Determinant as an alternating $n$-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 ${\cal D}$-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