Introduction to Linear Algebra
Complex Variables
- 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
- Linear transformations
- Affine maps
- Exercises
- Answers
Series and Limits
- Introduction
- Defunition of derivative
- Geometric meaning
- Properties Matrix operators
- Determinants
- Cofactors
- Cramer's rule
- Equivalent matrices
- Elimination: A = L U
- PLU factorization
- Reflection
- Givens rotation
- Special matrices
- Exercises
- Answers
Differentiation
- Introduction
- Defunition of derivative
- Geometric meaning
- Properties Compositions
- Derivative of inverse
- Power rule
- Right and Left-hand derivatives
- Higher order derivatives
- Infinitesimals
- Differentials
- Extrema of a real-valued functions
- Fermat's and Rolle's theorems
- Lagrange theorem
- de l'Hospital's rule
- Dual transformations
- Direct sums
- Quotient spaces
- Rank
- Solving A x = b
- Exercises
- Answers
Integration
- Introduction
- Riemann integral
- Area and integral
- Properties of integration
- Primitive functions
- Fundamental theorem of calculus
- Integration by parts
- Indefinite integrals
- Improper integrals
- Lebesgue integral
- Properties
- Double integral
- Polar coordinates
- Self-adjoint operators
Vector Calculus
- Introduction
- Vector fields
- Paths
- Line integrals
- Exercises
- Answers
- Inner product
- Norm and distance
- Matrix norms
- Dual norms
- Dual transformations
- Examples of transformations
Tensors
- Introduction
- LU-decomposition
- QR-decomposition
- Cholesky decomposition
- Schur decomposition
- Positive matrices
- Roots
- Polar factorization
- Spectral decomposition
- Singular values
- SVD <
- Pseudoinverse
- Exercises
- Answers
Calculus of Variations
- Introduction
- 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
Classical mechanics
- Introduction
- Similar matrices
- Diagonalization
- Sylvester formula
- The Resolvent method
- Polynomial interpolation
- Positive matrices
- Roots <
- Pseudoinverse
- Exercises
- Answers
Quantum mechanics
- Introduction
- 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
- Differential forms
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
This Book is licensed under Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License
Differential Operators
- Apostol, T.M., Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability, Wiley; 2nd edition, 1991; ISBN-13: 978-0471000075.
- Apostol, T.M., Mathematical Analysis, Pearson; 2nd edition, 1974; ISBN-13: 978-0201002881
- Fichtenholz, G.M., Fundamentals of Mathematical Analysis: International Series of Monographs in Pure and Applied Mathematics, Volume 2, Pergamon, 2013; ISBN-13 : 978-1483121710.
- Hubbard, J.H. and Hubbard, B.B., Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Matrix Editions; 5th edition, 2015; ISBN-13: 978-0971576681
- Kaplan, W., Advanced Calculus, Pearson; 5th edition, 2002; ISBN-13: 978-0201799378
- Grisvard, P., Elliptic Problems in Nonsmooth Domains, SIAM, 2011.