This consists of the elementary aspects of linear algebra which depend mainly on row operations involving elementary manipulations of matrices. The field of scalars is typically the field of complex numbers.

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• Preface

1. Preliminaries
   1. Sets And Set Notation
   2. Functions
   3. The Number Line And Algebra Of The Real Numbers
   4. Ordered fields
   5. The Complex Numbers
   6. The Fundamental Theorem Of Algebra
   7. Exercises
   8. Completeness of R
   9. Well Ordering And Archimedean Property
   10. Division
   11. Systems Of Equations
   12. Exercises
   13. Fn
   14. Algebra in Fn
   15. Exercises
   16. The Inner Product In Fn
   17. What Is Linear Algebra?
   18. Exercises
2. Linear Transformations
   1. Matrices
   2. Exercises
   3. Linear Transformations
   4. Some Geometrically Dened Linear Transformations
   5. The Null Space Of A Linear Transformation
   6. Subspaces And Spans
   7. An Application To Matrices
   8. Matrices And Calculus
   9. Exercises
3. Determinants
   1. Basic Techniques And Properties
   2. Exercises
   3. The Mathematical Theory Of Determinants
   4. The Cayley Hamilton Theorem
   5. Block Multiplication Of Matrices
   6. Exercises
4. Row Operations
   1. Elementary Matrices
   2. The Rank Of A Matrix
   3. The Row Reduced Echelon Form
   4. Rank And Existence Of Solutions To Linear Systems
   5. Fredholm Alternative
   6. Exercises
5. Some Factorizations
   1. LU Factorization
   2. Finding An LU Factorization
   3. Solving Linear Systems Using An LU Factorization
   4. The PLU Factorization
   5. Justification For The Multiplier Method
   6. Existence For The PLU Factorization
   7. The QR Factorization
   8. Exercises
6. Spectral Theory
   1. Eigenvalues And Eigenvectors Of A Matrix
   2. Some Applications Of Eigenvalues And Eigenvectors
   3. Exercises
   4. Schur’s Theorem
   5. Trace And Determinant
   6. Quadratic Forms
   7. Second Derivative Test
   8. The Estimation Of Eigenvalues
   9. Advanced Theorems
   10. Exercises
   11. Cauchy’s Interlacing Theorem for Eigenvalues