Introduction to Linear Algebra

• Linear Algebra Software

Systems of Linear Equations

• Introduction
• Linear Systems
• Vectors
• Linear combinations
• Matrices
• Planes in ℝ³
• Row operations
• Gaussian elimination
• Reduced Row-Echelon Form
• Equation A x = b
• Sensitivity of solutions
• Plane transformations
• Exercises
• Answers

Matrix Algebra

• Introduction
• Manipulation of matrices
• Matrix transformations
• Matrix norms
• 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
• Dimension
• Coordinate systems
• Change of basis
• Linear transformation
• Subspaces
• Intersections
• Direct Sums
• Matrix Spaces
• Row space
• Range or Column Space
• Rank
• Null Spaces or Kernels
• Dimension Theorems
• Four Subspaces
• Solving A x = b
• Exercises
• Answers

Eigenvalues, Eigenvectors

• Introduction
• Characteristic Polynomials
• Algebraic and Geometric Multiplicities
• Minimal Polynomials
• Eigenspaces
• Where are Eigenvalues?
• Eigenvalues of AB and BA
• Generalized Eigenvectors

Euclidean Vector Spaces

• Introduction
• Dot product
• Inner product
• Vector products
• Norm and distance
• Matrix norms
• 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
• Similar matrices
• Diagonalization
• Sylvester Formula
• The Resolvent Method
• Polynomial Interpolation
• Positive Matrices
• Roots
• Polar Factorization
• Spectral Decomposition
• SVD
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• Pseudoinverse
• Exercises
• Answers

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

Glossary

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Reference

• Free Materials
• Books
• GFDL License