Introduction to Linear Algebra
Systems of Linear Equations
- Introduction
- Linear systems
- Vectors
- Linear combinations
- Matrices
- Planes in ℝ³
- Reduced Row-Echelon Form
- Equation A x = b
- Sensitivity of solutions
- Linear independence
- Plane transformations
- Space transformations
- Rotations
- Linear transformations
- Affine maps
- Exercises
- Answers
Matrix Algebra
- Introduction
- Manipulation of matrices
- Partitioned matrices
- Block matrices Matrix operators
- Determinants
- Cofactors
- Cramer's rule
- Equivalent matrices
- Elimination: A = L U
- PLU factorization
- Reflection
- Givens rotation
- Special matrices
- Exercises
- Answers
Vector Spaces
- Introduction
- Motivation
- Vector Spaces
- Bases
- Dimension
- Coordinate systems
- Change of basis
- Linear transformations
- Matrix transformations
- Compositions
- Isomorphisms
- Dual transformations
- Direct sums
- Quotient spaces
- Vector products
- Cross products
- Rank
- Solving A x = b
- Exercises
- Answers
Eigenvalues, Eigenvectors
- Introduction
- Characteristic polynomials
- Companion matrix
- Algebraic and geometric multiplicities
- Minimal polynomials
- Eigenspaces
- Where are eigenvalues?
- Eigenvalues of A B and B A
- Generalized Eigenvectors
- Similarity
- Diagonalizability
- Self-adjoint operators
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
- Symmetric matrices
- LU-decomposition
- Sylvester Formula
- Cholesky decomposition
- Schur decomposition
- Positive matrices
- Roots
- Polar factorization
- Spectral decomposition
- Singular values
- SVD <
- Pseudoinverse
- Exercises
- Answers
Applications
- GPS problem
- Poisson equation
- Graph theory
- Error correcting codes
- Electric circuits
- Markov chains
- Cryptography
- Wave-length transfer matrix
- Computer graphics
- Linear Programming
- Hill's determinant
- Fibonacci matrices
- Discrete dynamic systems
- Discrete Fourier transform
- Fast Fourier transform
- Curve fitting
Functions of Matrices
- Introduction
- Diagonalization
- Sylvester formula
- The Resolvent method
- Polynomial interpolation
- Positive matrices
- Roots <
- Pseudoinverse
- Exercises
- Answers
Miscellany
- Circles along curves
- TNB frames
- Tensors
- Tensors in ℝ³
- Tensors & Mechanics
- Differential forms
- Calculus
- 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
Glossary
Reference
This Book is licensed under Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License
Inverse Matrices
Gaussian elimination
Gaussian elimination
- Axier, S., Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. 2015, ISBN 978-3-319-11079-0.
- Beezer, R.A., A First Course in Linear Algebra, 2017.
- Dillon, M., Linear Algebra, Vector Spaces, and Linear Transformations, American Mathematical Society, Providence, RI, 2023.