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linear extension (Definition)

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

$\displaystyle m=\sum\limits _{b\in B}m_b b,$    

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
$\displaystyle m \mapsto\sum\limits _{b\in B}m_bf_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

$\displaystyle \varphi_2\colon$ $\displaystyle M^2\to N$ $\displaystyle (m,n)$ $\displaystyle \mapsto\sum\limits _{b\in B}\sum\limits _{c\in B}m_bn_cf_2(b,c),$    

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

$\displaystyle \varphi_n\colon$ $\displaystyle M^n\to N$ $\displaystyle m$ $\displaystyle \mapsto\sum\limits _{b\in B^n}m_bf_n(b)$    

quite compactly using multi-index notation: $ m_b=\prod\limits _{k=1}^nm_{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.



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"linear extension" is owned by GrafZahl. [ full author list (2) ]
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See Also: basis, basis

Also defines:  bilinear extension, multilinear extension, $n$-linear extension
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Cross-references: similar, images, multilinear, multi-index notation, integer, positive, bilinear map, generated by, functor, free module, forgetful functor, adjoint functors, category of sets, category, category theory, terms, restriction, homomorphism, map, basis, commutative ring
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This is version 4 of linear extension, born on 2005-07-19, modified 2007-04-05.
Object id is 7240, canonical name is LinearExtension.
Accessed 3396 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
Pending Errata and Addenda
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