Contents
Introduction 1
Chapter 1. Systems of Linear Equations
Two Linear Equations with Two Variables
Basic Notations for Systems of Linear Equations
Elementary Transformations of Systems of Linear Equations and
Elementary Row Transformations of Matrices
m*ethodes for Solving Homogeneous and Nonhomogeneous Systems of
Linear Equations
Two Problems
Chapter 2. Vector Spaces
Fields
Vector Spaces

Linear Combinations and Basis of a Vector Space
Linear Dependence and Existence of a Basis
The Rank of a Finite System of Vectors
. The Dimension of a Vector Space
Direct Sum and Linear Complements
Row and Column Rank of a Matrix
. Application to Systems of Linear Equations
Chapter 3. Linear Maps
Denition and Simple Properties
Isomorphisms and Isomorphism of Vector Spaces
Dimension Formula for Linear Maps
The Vector Space HomF (V;W)
Linear Maps and Matrices
The Matrix Product
The Matrix Des c r i p tion of EndF (V )
Isomorphisms (Again)
Change of b*ases
e*quivalence and Similarity of Matrices
The General Linear Group
Application to Systems of Linear Equations (Again)
Chapter 4. Determinants
The Concept of a Determinant Function
Proof of Existence and Expansion of a Determinant with Respect to a
Row





. Elementary Properties of a Determinant
. The Leibniz Formula for Determinants
Appendix A. Some Terminology about Sets and Maps
Sets
iii
Maps
Appendix B. Fields with Positive Characteristic
Appendix C. Zorns Lemma and the Existence of a Basis
Appendix D. A Summary of Some Algebraic Structures.
Appendix E. About the Concept of a Rank
Index
Bibliography