linear extension

Let R be a commutative ring, M a free R-module, B a basis of M, and N a further R-module. Each element mM then has a unique representation


where mbR for all bB, and only finitely many mb are non-zero. Given a set map f1:BN we may therefore define the R-module homomorphismMathworldPlanetmath φ1:MN, called the linear extension of f1, such that


The map φ1 is the unique homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from M to N whose restrictionPlanetmathPlanetmathPlanetmath to B is f1.

The above observation has a convenient reformulation in terms of category theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath. Let 𝖱𝖬𝗈𝖽 denote the categoryMathworldPlanetmath of R-modules, and 𝖲𝖾𝗍 the category of sets. Consider the adjoint functorsMathworldPlanetmathPlanetmathPlanetmath U:𝖱𝖬𝗈𝖽𝖲𝖾𝗍, the forgetful functorMathworldPlanetmathPlanetmath that maps an R-module to its underlying set, and F:𝖲𝖾𝗍𝖱𝖬𝗈𝖽, the free moduleMathworldPlanetmathPlanetmath functorMathworldPlanetmath that maps a set to the free R-module generated by that set. To say that U is right-adjoint to F is the same as saying that every set map from B to U(N), the set underlying N, corresponds naturally and bijectively to an R-module homomorphism from M=F(B) to N.

Similarly, given a map f2:B2N, we may define the bilinear extension

φ2: M2N (m,n) bBcBmbncf2(b,c),

which is the unique bilinear map from M2 to N whose restriction to B2 is f2.

Generally, for any positive integer n and a map fn:BnN, we may define the n-linear extension

φn: MnN m bBnmbfn(b)

quite compactly using multi-index notation: mb=k=1nmk,bk.


The notion of linear extension is typically used as a manner-of-speaking. Thus, when a multilinear map is defined explicitly in a mathematical text, the images of the basis elements are given accompanied by the phrase “by multilinear extension” or similar.

Title linear extension
Canonical name LinearExtension
Date of creation 2013-03-22 15:24:06
Last modified on 2013-03-22 15:24:06
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 7
Author GrafZahl (9234)
Entry type Definition
Classification msc 15-00
Related topic basis
Related topic Basis
Defines bilinear extension
Defines multilinear extension
Defines n-linear extension