Indexed family

Last updated

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).

Contents

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.

Mathematical statement

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 indexedfamily of elements inXindexed byI, which is denoted by or simply (xi), 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 collection is used instead of set since a family can have the identical element multiple times (while a set is a collection of unordered and different objects) as long as each identical element is indexed differently.

Functions and families are formally equivalent, as any function f with a domain I induces a family (f(i))iI. 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 ij such that xi = xj. Thus, , where |A| denotes the cardinality of the set A. It means that a family can have the same element multiple times as long as these are indexed differently, and this is a difference between indexed families and sets. For example, , where the index set is the set of natural numbers.

Any set X gives rise to a family (xx)xX as X being indexed by itself. Thus any set naturally becomes a family. For any family (Ai)iI there is the set of all elements {Ai | iI}, but this does not carry any information about multiple containment of the same element (indexed differently) or the structure given by I. Hence, by using a set instead of the family, some information might be lost.

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).

Examples

Indexed vectors

For example, consider the following sentence:

The vectors v1, …, vn are linearly independent.

Here (vi)i ∈ {1, …, n} denotes a family of vectors. The i-th vector vi 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 v1 = v2 = (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).

Matrices

Suppose a text states the following:

A square matrix A is invertible, if and only if the rows of A are 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.)

Other examples

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

Operations on indexed families

Index sets are often used in sums and other similar operations. For example, if (ai)iI is an indexed family of numbers, the sum of all those numbers is denoted by

When (Ai)iI is a family of sets, the union of all those sets is denoted by

Likewise for intersections and Cartesian products.

Indexed subfamily

An indexed family (Bi)iJ is a subfamily of an indexed family (Ai)iI, if and only if J is a subset of I and Bi = Ai holds for all i in J.

Usage in category theory

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 J, and related by morphisms depending on two indices.

See also

Related Research Articles

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.

Basis (linear algebra) Subset of a vector space that allows defining coordinates

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.

Sequence Finite or infinite ordered list of elements

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.

Vector space Basic algebraic structure of linear algebra

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.

Linear subspace

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.

Function (mathematics) Mapping that associates a single output value to each input

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.

Change of basis Change of coordinates for a vector space

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.

Matrix (mathematics) Two-dimensional array of numbers with specific operations

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.

Cartesian product Mathematical set formed from two given sets

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 .

References