Introduction to Linear Algebra

• Linear Algebra Software

Systems of Linear Equations

• Introduction
• Linear Systems
• Vectors
• Norm and distance
• Dot product
• Linear combinations
• Matrices
• Manipulation of matrices
• Matrix norms
• Gaussian elimination
• Reduced Row-Echelon Form
• Equation A x = b
• Sensitivity of solutions
• Plane transformations
• Exercises
• Answers

Matrix Algebra

• Introduction
• Matrix transformations
• Determinants
• Cofactors
• Cramer's rule
• Partitioned matrices
• Elementary Matrices
• Inverse matrices
• Elimination: A = LU
• PLU factorization
• Reflection
• Givens rotation
• Special matrices
• Exercises
• Answers

Vector Spaces

• Introduction
• Vector Spaces
• Linear Independence
• Bases and Dimensions
• Coordinates and basis
• Subspaces
• Matrix Spaces
• Row space
• Range or Column Space
• Rank
• Null Spaces or Kernels
• Dimension Theorems
• Four Subspaces
• Solving A x = b
• Eigenspaces
• Where are Eigenvalues?
• Eigenvalues of AB and BA
• Generalized Eigenvectors
• Direct Sums
• Exercises
• Answers

Euclidean Vector Spaces

• Introduction
• Orthogonality
• Orthogonal Sets
• Self-adjoint Matrices
• Cholesky decomposition
• Unitary Matrices
• Projection Operators
• Gram--Schmidt Process
• QR-decomposition
• Least Square Approximation
• Quadratic Forms
• Exercises
• Answers

Functions of Matrices

• Introduction
• Characteristic Polynomials
• Algebraic and Geometric Multiplicities
• Minimal Polynomials
• Diagonalization
• Sylvester Formula
• The Resolvent Method
• Polynomial Interpolation
• Positive Matrices
• Roots
• Polar Factorization
• Spectral Decomposition
• SVD
• Exercises
• Answers

Numerical Methods

• Computer solves Systems of Linear Equations
• Location of Eigenvalues
• Power Method
• Singular Value Decomposition
• Iterative Method
• Similarity and Diagonalization

Applications

• GPS Problem
• Graph Theory
• Error Correcting Codes
• Electric Circuits
• Markov Chains
• Cryptography
• Wave-length Transfer Matrix
• Computer Graphics
• Linear Programming
• Hill's Determinant
• Fibonacci Matrices
• Discrete Fourier Transform
• Fast Fourier Transform

Miscellany

• Vector Representations
• Matrix Representations
• Change of Basis
• Orthonormal Diagonalization
• Generalized Inverse

Preliminaries

• Complex Number Operations
• Sets
• Polynomials
• Polynomials and Matrices

Glossary

• ABCDEFGHIJKLM
• NOPQRSTUVWXZ

Reference

• Free Materials
• Books
• GFDL License