linear algebra

Linear algebra is the branch of mathematics devoted to the theory of linear structureMathworldPlanetmath. The axiomatic treatment of linear structure is based on the notions of a linear spacePlanetmathPlanetmath (more commonly known as a vector space), and a linear mapping. Broadly speaking, there are two fundamental questions considered by linear algebra:

From the geometric point of view, “linear” is synonymous with “straight”, and consequently linear algebra can be regarded as the branch of mathematics dealing with lines and planes, as well as with transformationsMathworldPlanetmath of space that preserve “straightness”, e.g. rotationsMathworldPlanetmath and reflections. The two fundamental questions, in geometric terms, deal with

Linearity is a very basic notion, and consequently linear algebra has applications in numerous areas of mathematics, science, and engineering. Diverse disciplines, such as differential equations, differential geometry, the theory of relativity, quantum mechanics, electrical circuitsMathworldPlanetmath, computer graphics, and information theory benefit from the notions and techniques of linear algebra.

Euclidean geometryMathworldPlanetmath is related to a specialized branch of linear algebra that deals with linear measurement. Here the relevant notions are length and angle. A typical question is the determination of lines perpendicularMathworldPlanetmathPlanetmathPlanetmath to a given plane. A somewhat less specialized branch deals with affine structure, where the key notion is that of area and volume. Here determinantsMathworldPlanetmath play an essential role.

Yet another branch of linear algebra is concerned with computation, algorithms, and numerical approximation. Important examples of such techniques include: Gaussian eliminationMathworldPlanetmath, the method of least squares, LU factorization, QR decompositionMathworldPlanetmath, Gram-Schmidt orthogonalizationPlanetmathPlanetmath, singular value decompositionMathworldPlanetmath, and a number of iterative algorithms for the calculation of eigenvalues and eigenvectors.

The following subject outline surveys key topics in linear algebra.

  1. 1.

    Linear structure.

    1. (a)

      Introduction: systems of linear equations, Gaussian elimination, matrices, matrix operations.

    2. (b)

      Foundations: fields and vector spaces, subspacePlanetmathPlanetmath, linear independencePlanetmathPlanetmath, basis, ordered basis, dimensionPlanetmathPlanetmath, direct sumMathworldPlanetmathPlanetmath decomposition.

    3. (c)

      Linear mappings: linearity axioms, kernels and images, injectivity, surjectivity, bijectionsMathworldPlanetmath, compositionsMathworldPlanetmathPlanetmath, inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, matrix representations, change of bases, conjugationMathworldPlanetmath, similarityMathworldPlanetmath.

  2. 2.

    Affine structure.

    1. (a)

      Determinants: characterizing properties, cofactor expansion, permutationsMathworldPlanetmath, Cramer’s rule, classical adjoint.

    2. (b)

      Geometric aspects: Euclidean volume, orientation, equiaffine transformations, determinants as geometric invariantsMathworldPlanetmath of linear transformations.

  3. 3.

    Diagonalization and Decomposition.

    1. (a)
    2. (b)

      Obstructions: imaginary eigenvalues, nilpotent transformations, classification of 2-dimensional real transformations.

    3. (c)
  4. 4.


    1. (a)

      Foundations: vector space dual, bilinearity, bilinear transposeMathworldPlanetmath, Gram-Schmidt orthogonalization.

    2. (b)
    3. (c)

      Tensor algebra: tensor product, contractionPlanetmathPlanetmath, invariants of linear transformations, symmetry operations.

  5. 5.

    Euclidean and Hermitian structure.

    1. (a)

      Foundations: inner product axioms, the adjointPlanetmathPlanetmath operationMathworldPlanetmath, symmetricMathworldPlanetmathPlanetmathPlanetmathPlanetmath transformations, skew-symmetric transformations, self-adjoint transformations, normal transformations.

    2. (b)

      Spectral theorem: diagonalization of self-adjoint transformations, diagonalization of quadratic forms.

  6. 6.

    Computational and numerical methods.

    1. (a)

      Linear problems: LU-factorization, QR decomposition, least squares, Householder transformations.

    2. (b)

      Eigenvalue problems: singular value decomposition, Gauss and Jacobi-Siedel iterative algorithms.

Title linear algebra
Canonical name LinearAlgebra
Date of creation 2013-03-22 12:26:16
Last modified on 2013-03-22 12:26:16
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Topic
Classification msc 15-00
Classification msc 15-01