This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms.

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This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms. At the end, the more abstract notions of vector spaces and linear transformations on vector spaces are presented. This is intended to be a first course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra.

- Some Prerequisite Topics
- Sets And Set Notation
- Functions
- Graphs Of Functions
- The Complex Numbers
- Polar Form Of Complex Numbers
- Roots Of Complex Numbers
- The Quadratic Formula
- Exercises

- Fn
- Algebra in Fn
- Geometric Meaning Of Vectors
- Geometric Meaning Of Vector Addition
- Distance Between Points In Rn Length Of A Vector
- Geometric Meaning Of Scalar Multiplication
- Exercises
- Vectors And Physics
- Exercises

- Vector Products
- The Dot Product
- The Geometric Significance Of The Dot Product
- Exercises
- The Cross Product
- The Vector Identity Machine
- Exercises

- Systems Of Equations
- Systems Of Equations, Geometry
- Systems Of Equations, Algebraic Procedures
- Exercises

- Matrices
- Matrix Arithmetic
- Exercises

- Determinants
- Basic Techniques And Properties
- Applications
- Exercises

- The Mathematical Theory Of Determinants
- The Function sgnn
- The Determinant
- The Cayley Hamilton Theorem

- Rank Of A Matrix
- Elementary Matrices
- THE Row Reduced Echelon Form Of A Matrix
- The Rank Of A Matrix
- Fredholm Alternative
- Exercises

- Linear Transformations
- Linear Transformations
- Constructing The Matrix Of A Linear Transformation
- Exercises

- The LU Factorization
- Definition Of An LU factorization
- Finding An LU Factorization By Inspection
- Using Multipliers To Find An LU Factorization
- Solving Systems Using The LU Factorization
- Justification For The Multiplier Method
- The PLU Factorization
- The QR Factorization
- Exercises

- Linear Programming
- Simple Geometric Considerations
- The Simplex Tableau
- The Simplex Algorithm
- Finding A Basic Feasible Solution
- Duality
- Exercises

- Spectral Theory
- Eigenvalues And Eigenvectors Of A Matrix
- Some Applications Of Eigenvalues And Eigenvectors
- The Estimation Of Eigenvalues
- Exercises

- Matrices And The Inner Product
- Symmetric And Orthogonal Matrices
- Fundamental Theory And Generalizations
- Least Square Approximation
- The Right Polar Factorization
- The Singular Value Decomposition
- Approximation In The Frobenius Norm
- Moore Penrose Inverse
- Exercises

- Numerical Methods For Solving Linear Systems
- Iterative Methods For Linear Systems
- The Operator Norm
- The Condition Number
- Exercises

- Numerical Methods For Solving The Eigenvalue Problem
- The Power Method For Eigenvalues
- The Shifted Inverse Power Method
- The Rayleigh Quotient
- The QR Algorithm
- Exercises
- Vector Spaces
- Algebraic Considerations
- Exercises
- Vector Spaces And Fields
- Exercises
- Inner Product Spaces
- Exercises

- Linear Transformations
- Matrix Multiplication As A Linear Transformation
- L(V,W) As A Vector Space
- Eigenvalues And Eigenvectors Of Linear Transformations
- Block Diagonal Matrices
- The Matrix Of A Linear Transformation
- Exercises

- The Jordan Canonical Form*
- The Fundamental Theorem Of Algebra
- Answers To Selected Exercises

Excellent, very comprehensive, and well written. Includes problem sets and answers. Portions would be a great primer for a university class, and is a great reference for theory.

June 4, 2015 at 12:43 PM