# 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

$$m=\sum _{b\in B}{m}_{b}b,$$ |

where ${m}_{b}\in R$ for all $b\in B$, and only finitely many ${m}_{b}$ are
non-zero. Given a set map ${f}_{1}:B\to N$ we may therefore define the $R$-module homomorphism^{} ${\phi}_{1}:M\to N$, called the *linear extension* of ${f}_{1}$, such that

$$m\mapsto \sum _{b\in B}{m}_{b}{f}_{1}(b).$$ |

The map ${\phi}_{1}$ 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 $\mathrm{\U0001d5b1\U0001d5ac\U0001d5c8\U0001d5bd}$ denote the category^{} of $R$-modules, and $\mathrm{\U0001d5b2\U0001d5be\U0001d5cd}$ the category of sets. Consider the adjoint functors^{} $U:\mathrm{\U0001d5b1\U0001d5ac\U0001d5c8\U0001d5bd}\to \mathrm{\U0001d5b2\U0001d5be\U0001d5cd}$, the forgetful functor^{} that maps an $R$-module to its underlying set, and $F:\mathrm{\U0001d5b2\U0001d5be\U0001d5cd}\to \mathrm{\U0001d5b1\U0001d5ac\U0001d5c8\U0001d5bd}$,
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 ${f}_{2}:{B}^{2}\to N$, we may define the
*bilinear extension*

${\phi}_{2}:$ | ${M}^{2}\to N$ | $(m,n)$ | $\mapsto {\displaystyle \sum _{b\in B}}{\displaystyle \sum _{c\in B}}{m}_{b}{n}_{c}{f}_{2}(b,c),$ |

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 ${f}_{n}:{B}^{n}\to N$, we may define the *$n$-linear extension*

${\phi}_{n}:$ | ${M}^{n}\to N$ | $m$ | $\mapsto {\displaystyle \sum _{b\in {B}^{n}}}{m}_{b}{f}_{n}(b)$ |

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

## 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 |