concepts in linear algebra

The aim of this entry is to present a list of the key objects and operators used in linear algebraMathworldPlanetmath. Each entry in the list links (or will link in the future) to the corresponding PlanetMath entry where the object is presented in greater detail. For convenience, this list also presents the encouraged notation to use (at PlanetMath) for these objects.

Some of this notation is simply an example of more general notation, either notation in set theoryMathworldPlanetmath or notation for functions. Some notation is also standard from category theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Suppose V is a vector spaceMathworldPlanetmath over a field K. Where the field K is clear from context it is sometimes eliminated from the notation. Let L be a linear operator, or linear transformation, from V to W, and E be an endomorphismPlanetmathPlanetmathPlanetmath of V.

  • (V,W), the set of linear transformations between vector spaces V and W (also HomK(V,W)),

  • basis of a vector space and the matrix associated with the basis,

  • dimKV, dimensionPlanetmathPlanetmath ( of V,

  • span{e1,,en} vector space spanned by vectors {ei} (note that the list of vectors need not be finite). Some other notations are (e1,e2,,en) or e1,e2,,en (do not confuse last one with similarPlanetmathPlanetmath inner product notation),

  • detE, determinantMathworldPlanetmath of a linear operator,

  • trE, trace of a linear operator (also traceE),

  • imL, image of a linear operator (also imgL and L(V)),

  • kerL, kernel of a linear operator,

  • for sets A,B and a point xV, the expressions A+B, A-B, A+x are the Minkowski sumsPlanetmathPlanetmath ( (not especially if A and B are subspacesPlanetmathPlanetmath, then their sum is a subspace, but the result may not be a direct sumMathworldPlanetmathPlanetmathPlanetmathPlanetmath),

  • V, dual spacePlanetmathPlanetmath of a vector space V (also V or HomK(V,K)),

  • A, adjoint operator of a linear operator,

  • VW, direct sum of vector spaces V and W (both internal end external),

  • VKW, tensor productPlanetmathPlanetmathPlanetmath of V and W, and

  • VW, antisymmetrized tensor product (also called the wedge productMathworldPlanetmath),

  • generalizationsPlanetmathPlanetmath of vector spaces over a field to vector spaces over a division ring to modules over a ring.

Title concepts in linear algebra
Canonical name ConceptsInLinearAlgebra
Date of creation 2013-03-22 14:13:30
Last modified on 2013-03-22 14:13:30
Owner matte (1858)
Last modified by matte (1858)
Numerical id 13
Author matte (1858)
Entry type Definition
Classification msc 16-00
Classification msc 13-00
Classification msc 20-00
Classification msc 15-00