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 $m\in M$ then has a unique representation

m=\sum\limits_{b\in B}m_{b}b,

where $m_{b}\in R$ for all $b\in B$ , and only finitely many $m_{b}$ are non-zero. Given a set map $f_{1}\colon B\to N$ we may therefore define the $R$ -module homomorphism $\varphi_{1}\colon M\to N$ , called the linear extension of $f_{1}$ , such that

m\mapsto\sum\limits_{b\in B}m_{b}f_{1}(b).

The map $\varphi_{1}$ is the unique homomorphism from $M$ to $N$ whose restriction to $B$ is $f_{1}$ .

The above observation has a convenient reformulation in terms of category theory. Let $\mathsf{RMod}$ denote the category of $R$ -modules, and $\mathsf{Set}$ the category of sets. Consider the adjoint functors $U\colon\mathsf{RMod}\to\mathsf{Set}$ , the forgetful functor that maps an $R$ -module to its underlying set, and $F\colon\mathsf{Set}\to\mathsf{RMod}$ , the free module functor 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 $f_{2}\colon B^{2}\to N$ , we may define the bilinear extension

\displaystyle\varphi_{2}\colon

\displaystyle M^{2}\to N

\displaystyle(m,n)

\displaystyle\mapsto\sum\limits_{b\in B}\sum\limits_{c\in B}m_{b}n_{c}f_{2}(b,% c),

which is the unique bilinear map from $M^{2}$ to $N$ whose restriction to $B^{2}$ is $f_{2}$ .

Generally, for any positive integer $n$ and a map $f_{n}\colon B^{n}\to N$ , we may define the $n$ -linear extension

\displaystyle\varphi_{n}\colon

\displaystyle M^{n}\to N

\displaystyle m

\displaystyle\mapsto\sum\limits_{b\in B^{n}}m_{b}f_{n}(b)

quite compactly using multi-index notation: $m_{b}=\prod\limits_{k=1}^{n}m_{k,b_{k}}$ .

Usage

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