Tuples, vectors, matrices/R/Sets/Introduction/Section

Important product sets are ${}\mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R}$ and ${}\mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R}$. The ordering of the elements is essential. In general, for a set ${}M$ and some ${}n\in \mathbb {N}$, we denote the ${}n$-th fold product set of ${}M$ with itself as

${}M^{n}=\underbrace {M\times \cdots \times M} _{n{\text{ times}}}\,.$

The elements have the form

$\left(x_{1},x_{2},\ldots ,x_{n}\right),$

where every ${}x_{i}$ is from ${}M$. Such an ordered finite sequence of ${}n$ elements is also called an ${}n$-tuple over ${}M$. For ${}n=2$, it is called a pair, for ${}n=3$, it is called a triple. For

${}x=\left(x_{1},x_{2},\ldots ,x_{n}\right)\,,$

the element ${}x_{i}$ is called the ${}i$-th component or the ${}i$-th entry of the tuple. In this context, the ${}i$ is called the index of the tuple, and ${}\{1,\ldots ,n\}$ is called the index set of the tuple.

More generally, for every index set ${}I$, there exist ${}I$-tuples. In such an ${}I$-tuple, to every index ${}i\in I$ some mathematical object is assigned; the tuple is often written as ${}x_{i}$, ${}i\in I$. If all ${}x_{i}$ are from one set ${}M$, then we call this an ${}I$-tuple from ${}M$. For ${}I=\mathbb {N}$, we call this a sequence in ${}M$.

A finite index set can always be replaced by a set of the form ${}\{1,\ldots ,n\}$ (this procedure is called a numbering of the index set), but this is not always useful. If we start with the index set

${}I=\{1,\ldots ,n\}\,$

and if we are interested in a certain subset ${}J\subseteq I$, then it is natural to stick to the original notation from ${}I$ instead of introducing a new numbering ${}\{1,\ldots ,m\}$ for ${}J$. Quite often, there is a "natural“ index set for a certain problem that represents a part of the structure of the problem (and is easier to remember).

An ${}n$-tuple over a set ${}M$ of the form

$\left(a_{1},\,\ldots ,\,a_{n}\right)$

is also called a row tuple (of length ${}n$), and an ${}n$-tuple of the form

${\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}$

is called a column tuple. These are just two different ways to represent the tuple, but if the tuple represents some structure (like a vector, to which a matrix (see below) shall be applied), then this difference is relevant.

When ${}I$ and ${}J$ are two sets and ${}I\times J$ is their product set, then we can express an ${}I\times J$-tuple in ${}M$ as a "table“, that assigns to every pair ${}(i,j)$ an element ${}a_{ij}\in M$. In particular, for ${}I=\{1,\ldots ,m\}$ and ${}J=\{1,\ldots ,n\}$, we call an ${}I\times J$-tuple also an ${}m\times n$-matrix, and write this as

${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}.$

The row tuple

$\left(a_{i1},\,a_{i2},\,\ldots ,\,a_{in}\right)$

is called the ${}i$-th row of the matrix, and

${\begin{pmatrix}a_{1j}\\a_{2j}\\\vdots \\a_{mj}\end{pmatrix}}$

is called the ${}j$-th column of the matrix.