Rating:

( 24 )

208 pages

Language:

English

This consists of the elementary aspects of linear algebra which depend mainly on row operations involving elementary manipulations of matrices.

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Description

Content

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

- Preliminaries
- Sets And Set Notation
- Functions
- The Number Line And Algebra Of The Real Numbers
- Ordered fields
- The Complex Numbers
- The Fundamental Theorem Of Algebra
- Exercises
- Completeness of R
- Well Ordering And Archimedean Property
- Division
- Systems Of Equations
- Exercises
- Fn
- Algebra in Fn
- Exercises
- The Inner Product In Fn
- What Is Linear Algebra?
- Exercises

- Linear Transformations
- Matrices
- Exercises
- Linear Transformations
- Some Geometrically Dened Linear Transformations
- The Null Space Of A Linear Transformation
- Subspaces And Spans
- An Application To Matrices
- Matrices And Calculus
- Exercises

- Determinants
- Basic Techniques And Properties
- Exercises
- The Mathematical Theory Of Determinants
- The Cayley Hamilton Theorem
- Block Multiplication Of Matrices
- Exercises

- Row Operations
- Elementary Matrices
- The Rank Of A Matrix
- The Row Reduced Echelon Form
- Rank And Existence Of Solutions To Linear Systems
- Fredholm Alternative
- Exercises

- Some Factorizations
- LU Factorization
- Finding An LU Factorization
- Solving Linear Systems Using An LU Factorization
- The PLU Factorization
- Justification For The Multiplier Method
- Existence For The PLU Factorization
- The QR Factorization
- Exercises

- Spectral Theory
- Eigenvalues And Eigenvectors Of A Matrix
- Some Applications Of Eigenvalues And Eigenvectors
- Exercises
- Schur’s Theorem
- Trace And Determinant
- Quadratic Forms
- Second Derivative Test
- The Estimation Of Eigenvalues
- Advanced Theorems
- Exercises
- Cauchy’s Interlacing Theorem for Eigenvalues