Introduction to Linear Algebra
Systems of Linear Equations
- Introduction
- Linear systems
- Vectors
- Linear combinations
- Matrices
- Planes in ℝ³
- Equation A x = b
- Sensitivity of solutions
- Linear independence
- Plane transformations
- Space transformations
- Linear transformations
- Affine maps
- Exercises
- Answers
Matrix Algebra
- Introduction
- Manipulation of matrices
- Partitioned matrices
- Block matrices
- Matrix operators
- Determinants
- Cofactors
- Cramer's rule
- Chiò's method
- 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
- Linear transformations
- Change of basis
- Matrix transformations
- Compositions
- Isomorphisms
- Dual transformations
- Quotient spaces
- 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
- Exercises
- Answers
Euclidean Spaces
- Introduction
- Euclidean space
- Bilinear transformations
- Norm and distance
- Matrix norms
- Dual norms
- Dual transformations
- Examples of transformations
- Orthogonality
- Gram--Schmidt Process
- Orthogonal matrices
- Self-adjoint matrices
- Unitary matrices
- Projection operators
- QR-decomposition
- Least Square Approximation
- Quadratic forms
- Exercises
- Answers
Canonical forms
- Introduction
- 2D decomposition
- 3D decomposition
- Projectors
- Direct-sum decompositions
- Cyclic decompositions
- Symmetric matrices
- Pseudoinverse
- URV-decomposition
- LU-decomposition
- QR-decomposition
- Cholesky decomposition
- Schur decomposition
- Positive matrices
- Roots
- Polar factorization
- Spectral decomposition
- CUR decomposition
- Exercises
- Answers
Applications
- Introduction
- Circles along curves
- TNB frames
- GPS problem
- Coriolis acceleration
- Poisson equation
- Graph theory
- Error correcting codes
- Electric circuits
- FSA
- 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
- Answers
Miscellany
- Introduction
- Circles along curves
- TNB frames
- 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
In 1907, the Baltic German mathematician Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process. Schmidt claimed that his procedure was essentially the same as an earlier one published by the Danish actuary and mathematician J. P. Gram (1850--1916) in 1883. The Schmidt version was the first to become popular and widely used. An algorithm related to a modified version of the process appeared in an 1820 treatise by P. S. Laplace. Although related algorithms have been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthogonalization techniques.
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Gram--Schmidt process
The Gram-Schmidt process is an algorithm to transform an ordered set of vectors into an orthogonal or orthonormal set spanning the same subspace, that is generating the same collection of linear combinations.