Formal linear combination

In many areas of mathematics, especially in Algebra and its applications, such as Complex Analysis and Topology, it is often helpful to consider "linear combinations" of elements of a set that does not have a module structure. In this article, we aim to briefly explain how these linear combinations can be mathematically represented as actual linear combinations.

Let $M$ be a set and $R$ a ring (in many classical cases, $R$ is a field or the ring $\mathbb {Z}$ of integers).

Classical Definition

A formal linear combination of elements from $M$ over $R$ is a sum of the form $\sum _{i=1}^{n}r_{i}m_{i}$, where $r_{i}\in R$ and $m_{i}\in M$, with the $m_{i}$ being pairwise distinct. Two sums of this form are considered equal if the same elements of $M$ appear with the same coefficients (with a coefficient of $0$ meaning that the element does not appear). The set of these sums forms, with the operations

• $\sum _{i=1}^{n}r_{i}m_{i}+\sum _{i=1}^{n}s_{i}m_{i}:=\sum _{i=1}^{n}(r_{i}+s_{i})m_{i}$ (any two such sums can be written so that they contain the same elements of $M$; if necessary, terms of the form $0\cdot m$ with $m\in M$ can be added), with $r_{i},s_{i}\in R$, $m_{i}\in M$, $n\in \mathbb {N}$,

• $r\cdot \sum _{i=1}^{n}r_{i}m_{i}=\sum _{i=1}^{n}(rr_{i})m_{i}$ with $r_{i},r\in R$, $m_{i}\in M$, $n\in \mathbb {N}$,

a natural $R$-module (in the case of a field, this is a vector space; in the case of integers, an abelian group). This $R$-module is also called the free $R$-module over $M$.

The definition is somewhat unsatisfactory in that the elements of the free $R$-module are undefined sums $\sum _{i=1}^{n}r_{i}m_{i}$; note that we can only define the operations once we have the elements, but we need the operations to describe the elements. One could argue that the sums are merely "formal," but this is mathematically unsatisfactory since it is unclear what a formal sum actually is. The main idea of the precise definition is to assign the formal sums an exact meaning.

Preliminary Considerations for the Exact Definition

How can we assign the sum $s=\sum _{i=1}^{n}r_{i}m_{i}$ a mathematically precise object with the same meaning? The idea is to think about what we actually need from the sums. These are the pairs $(m_{i},r_{i})$. Remembering that the $m_{i}$ were pairwise distinct, we see that we are dealing with a map $m_{i}\mapsto r_{i}$. Extending this map to all of $R$ by assigning $0$ to other elements gives us a map $\phi _{s}\colon M\to R$ with the property that its so-called support $\mathrm {supp} (\phi _{s}):=\phi _{s}^{-1}[R\setminus {0}]$, i.e., the set of elements not mapped to zero, is finite. Conversely, if $\phi \colon M\to R$ is a map with finite support $\mathrm {supp} (\phi )={m_{1},\ldots ,m_{n}}$, we can assign it the formal sum $s_{\phi }:=\sum _{i=1}^{n}\phi (m_{i})m_{i}$. One can see that these constructions are inverses of each other, so formal sums correspond exactly to maps with finite support. We can thus define:

Definition

A formal linear combination of elements from $M$ over $R$ is a map $\phi \colon M\to R$ with

$|\mathrm {supp} (\phi )|<\infty$

The set of all such maps is denoted by $R^{(M)}$ and is an $R$-submodule of $R^{M}$. It is called the free $R$-module over $M$.

Notation

Let $\phi \in R^{(M)}$. Since sums are more convenient to work with than maps (as the sum compactly encodes all the information), we want to write $\phi$ as a sum. For $m\in M$, consider the map $\phi _{m}\colon M\to R$, which maps $m$ to $1$ and $M\setminus {m}$ to $0$. If ${m_{1},\ldots ,m_{n}}$ is a listing of the support of $\phi$ with pairwise distinct $m_{i}$, then $\phi =\sum _{i=1}^{n}\phi (m_{i})\phi _{m_{i}}$ (note that this is an actual sum in the $R$-module $R^{(M)}$ with pointwise operations). The notation is often further simplified by identifying the elements $m\in M$ with the map $\phi _{m}$, so $\phi _{m}$ is simply called $m$. Then $\phi =\sum _{i=1}^{n}\phi (m_{i})m_{i}$, allowing us to work with convenient sums while ensuring that everything rests on a solid mathematical foundation.

Translation and Version Control

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Formale Linearkombination Wikiversity source page] and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Formale Linearkombination - URL:

https://de.wikiversity.org/wiki/Formale Linearkombination

• Date: 12/17/2024