# 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 LinearExtension 2013-03-22 15:24:06 2013-03-22 15:24:06 GrafZahl (9234) GrafZahl (9234) 7 GrafZahl (9234) Definition msc 15-00 basis Basis bilinear extension multilinear extension $n$-linear extension