15-00-LinearExtension.tex

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\pmtitle{linear extension}
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\pmtype{Definition}
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\pmdefines{bilinear extension}
\pmdefines{multilinear extension}
\pmdefines{$n$-linear extension}

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\begin{document}
\PMlinkescapeword{extension}
\PMlinkescapeword{representation}
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
\begin{equation*}
m=\Sum_{b\in B}m_b b,
\end{equation*}
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 \emph{linear extension} of $f_1$, such that
\[
 m \mapsto\Sum_{b\in B}m_bf_1(b).
\]
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 $\RMod$ denote the category of $R$-modules, and $\Set$ the category of sets. Consider the adjoint functors $U\colon\RMod\to \Set$, the forgetful functor that maps an $R$-module to its underlying set, and $F \colon \Set \to \RMod$,
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
\emph{bilinear extension}
\begin{align*}
\varphi_2\colon&M^2\to N&(m,n)&\mapsto\Sum_{b\in B}\Sum_{c\in B}m_bn_cf_2(b,c),
\end{align*}
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 \emph{$n$-linear extension}
\begin{align*}
\varphi_n\colon&M^n\to N&m&\mapsto\Sum_{b\in B^n}m_bf_n(b)
\end{align*}
quite compactly using multi-index notation: $m_b=\Prod_{k=1}^nm_{k,b_k}$.

\subsection*{Usage}

The notion of linear extension is typically used as a
\emph{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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