In mathematics, a **family**, or **indexed family**, is informally a collection of objects, each associated with an index from some index set. For example, a *family of real numbers, indexed by the set of integers * is a collection of real numbers, where a given function selects one real number for each integer (possibly the same).

- Mathematical statement
- Examples
- Indexed vectors
- Matrices
- Other examples
- Operations on indexed families
- Indexed subfamily
- Usage in category theory
- See also
- References

More formally, an indexed family is a mathematical function together with its domain I and image X. Often the elements of the set X are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I is called the *index (set)* of the family, and X is the *indexed set*. Sequences are one type of families with the specific domains.

**Definition.** Let I and X be sets and f a function such that

where represents an element of I and as the image of under the function f is denoted as (e.g., is denoted as . The symbol is used to indicate that is an element of X.), then this establishes an **indexed****family of elements in**X**indexed by**I, which is denoted by or simply (*x _{i}*), when the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, the latter with the risk of mixing-up families with sets. Simply speaking, whenever index notation is used, the indexed objects form a (indexed) family as the collection of them. The term

Functions and families are formally equivalent, as any function *f* with a domain *I* induces a family (*f* (*i*))_{i∈I}. Being an element of a family is equivalent with being in the range of the corresponding function. In practice, however, a family is viewed as a collection, rather than a function. A family contains any element exactly once, if and only if the corresponding function is injective.

An indexed family can be turned into a set by considering the set , that is, the image of I under f. Since the mapping f is not required to be injective, there may exist with *i* ≠ *j* such that *x _{i}* =

Any set X gives rise to a family (*x _{x}*)

The index set I is not restricted to be countable, and a subset of a power set may be indexed, resulting in an **indexed family of sets**. Sequences are one type of families as a sequence is defined as a function with the specific domain (an interval of integers, the set of natural numbers, or the set of first n natural numbers, depending on what sequence is defined and what definition is used).

For example, consider the following sentence:

The vectors

v_{1}, …,v_{n}are linearly independent.

Here (*v*_{i})_{i ∈ {1, …, n}} denotes a family of vectors. The i-th vector *v*_{i} only makes sense with respect to this family, as sets are unordered so there is no i-th vector of a set. Furthermore, linear independence is defined as a property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family. For example, if we consider *n* = 2 and *v*_{1} = *v*_{2} = (1, 0) as the same vector, then the *set* of them consists of only one element (as a set is a collection of unordered distinct elements) and is linearly independent, but the family contains the same element twice (since indexed differently) and is linearly dependent (same vectors are linearly dependent).

Suppose a text states the following:

A square matrix

Ais invertible, if and only if the rows ofAare linearly independent.

As in the previous example, it is important that the rows of *A* are linearly independent as a family, not as a set. For example, consider the matrix

The *set* of the rows consists of a single element (1, 1) as a set is made of unique elements so it is linearly independent, but the matrix is not invertible as the matrix determinant is 0. On the other hands, the *family* of the rows contains two elements indexed differently such as the 1st row (1, 1) and the 2nd row (1,1) so it is linearly dependent. The statement is therefore correct if it refers to the family of rows, but wrong if it refers to the set of rows. (The statement is also correct when "the rows" is interpreted as referring to a multiset, in which the elements are also kept distinct but which lacks some of the structure of an indexed family.)

Let **n** be the finite set {1, 2, …, *n*}, where n is a positive integer.

- An ordered pair (2-tuple) is a family indexed by the set of two elements,
**2**= {1, 2}; each element of the ordered pair is indexed by each element of the set**2**. - An n-tuple is a family indexed by the set
**n**. - An infinite sequence is a family indexed by the natural numbers.
- A list is an n-tuple for an unspecified n, or an infinite sequence.
- An
*n*×*m*matrix is a family indexed by the Cartesian product**n**×**m**which elements are ordered pairs, e.g., (2, 5) indexing the matrix element at the 2nd row and the 5th column. - A net is a family indexed by a directed set.

Index sets are often used in sums and other similar operations. For example, if (*a*_{i})_{i∈I} is an indexed family of numbers, the sum of all those numbers is denoted by

When (*A*_{i})_{i∈I} is a family of sets, the union of all those sets is denoted by

Likewise for intersections and Cartesian products.

An indexed family (*B*_{i})_{i∈J} is a **subfamily** of an indexed family (*A*_{i})_{i∈I}, if and only if J is a subset of I and *B _{i}* =

The analogous concept in category theory is called a ** diagram **. A diagram is a functor giving rise to an indexed family of objects in a category * C*, indexed by another category

In computer science, an **array data structure**, or simply an **array**, is a data structure consisting of a collection of *elements*, each identified by at least one *array index* or *key*. An array is stored such that the position of each element can be computed from its index tuple by a mathematical formula. The simplest type of data structure is a linear array, also called one-dimensional array.

In mathematics, a **countable set** is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a finite set or a **countably infinite** set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number.

In mathematics, any vector space * has a corresponding ***dual vector space** consisting of all linear forms on *, together with the vector space structure of pointwise addition and scalar multiplication by constants.*

In mathematics, one can often define a **direct product** of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

In mathematics, a set B of vectors in a vector space *V* is called a **basis** if every element of *V* may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as **components** or **coordinates** of the vector with respect to B. The elements of a basis are called **basis vectors**.

In mathematics, a **sequence** is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the *length* of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.

A **vector space** is a set of objects called *vectors*, which may be added together and multiplied ("scaled") by numbers, called *scalars*. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector *axioms*. To specify that the scalars are real or complex numbers, the terms **real vector space** and **complex vector space** are often used.

In mathematics, and more specifically in linear algebra, a **linear subspace**, also known as a **vector subspace** is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a *subspace* when the context serves to distinguish it from other types of subspaces.

A **mathematical symbol** is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

In mathematics, a **function** is a binary relation between two sets that associates each element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

In mathematics and computer programming, **index notation** is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication, or when one is writing a computer program.

In mathematics, the **lexicographic** or **lexicographical order** is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set.

In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in general, different from the coordinate vector that represents v on the other basis. A **change of basis** consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis.

This page lists some **examples of vector spaces**. See vector space for the definitions of terms used on this page. See also: dimension, basis.

Many letters of the Latin alphabet, both capital and small, are used in mathematics, science, and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, or physical entities. Certain letters, when combined with special formatting, take on special meaning.

In mathematics, a **matrix** is a rectangular *array* or *table* of numbers, symbols, or expressions, arranged in *rows* and *columns*. For example, the dimension of the matrix below is 2 × 3, because there are two rows and three columns:

In computer science, an **array type** is a data type that represents a collection of *elements*, each selected by one or more indices that can be computed at run time during program execution. Such a collection is usually called an **array variable**, **array value**, or simply **array**. By analogy with the mathematical concepts vector and matrix, array types with one and two indices are often called **vector type** and **matrix type**, respectively. More generally, a multidimensional array type can be called a **tensor type**.

In mathematics and physics, a **vector** is an element of a vector space.

In mathematics, specifically set theory, the **Cartesian product** of two sets *A* and *B*, denoted *A* × *B*, is the set of all ordered pairs (*a*, *b*) where *a* is in *A* and *b* is in *B*. In terms of set-builder notation, that is

In mathematics, an **algebraic number field** is an extension field of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .

- Mathematical Society of Japan,
*Encyclopedic Dictionary of Mathematics*, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).

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