Introduction to Linear Algebra
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
- Linear Independence
- Plane transformations
- Linear transformations
- Exercises
- Answers
Matrix Algebra
- Introduction
- Manipulation of matrices
- Matrix transformations
- Block matrices
- 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
- Motivation
- Vector Spaces
- Bases
- Dimension
- Coordinate systems
- Change of basis
- Linear transformations
- Isomorphisms
- Dual spaces
- Dual transformations
- Subspaces
- Intersections
- Direct Sums
- Quotient spaces
- Vector products
- 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 Spaces
- Introduction
- Dot product
- Bilinear transformations
- Inner product
- Norm and distance
- Matrix norms
- Dual norms
- Dual transformations
- Orthogonality
- Gram--Schmidt Process
- Orthogonal Sets
- Self-adjoint Matrices
- Unitary Matrices
- Projection Operators
- QR-decomposition
- Least Square Approximation
- Quadratic Forms
- Exercises
- Answers
Matrix Decompositions
- Introduction
- LU-decomposition
- Sylvester Formula
- Cholesky decomposition
- Schur decomposition
- Jordan decomposition
- Positive Matrices
- Roots
- Polar Factorization
- Spectral Decomposition
- Singula values
- SVD <
- 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
- Differential forms
Functions of Matrices
- Introduction
- Similar matrices
- Diagonalization
- Sylvester Formula
- The Resolvent Method
- Polynomial Interpolation
- Positive Matrices
- Roots <
- Pseudoinverse
- Exercises
- Answers
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
Reference
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