Vector space in the context of Injective function

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday life.

Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers such as (x, y, z, w). For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled x, y, and z). It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of the simplest possible regular 4D objects, the tesseract, which is analogous to the 3D cube.

In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy.

An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called scalar multiplication between elements of the field (called scalars), and elements of the vector space (called vectors).

In mathematics, the absolute value or modulus of a real number ${\displaystyle x}$, denoted ${\displaystyle |x|}$, is the non-negative value of ${\displaystyle x}$ without regard to its sign. Namely, ${\displaystyle |x|=x}$ if ${\displaystyle x}$ is a positive number, and ${\displaystyle |x|=-x}$ if ${\displaystyle x}$ is negative (in which case negating ${\displaystyle x}$ makes ${\displaystyle -x}$ positive), and ${\displaystyle |0|=0}$. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement and possibly a support, formulated as a directed line segment. A vector is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B, and denoted by ${\textstyle {\stackrel {\longrightarrow }{AB}}.}$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means 'carrier'. It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

where ${\displaystyle V_{1},\ldots ,V_{n}}$ (${\displaystyle n\in \mathbb {Z} _{\geq 0}}$) and ${\displaystyle W}$ are vector spaces (or modules over a commutative ring), with the following property: for each ${\displaystyle i}$, if all of the variables but ${\displaystyle v_{i}}$ are held constant, then ${\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}$ is a linear function of ${\displaystyle v_{i}}$. One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the cross product likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of ${\displaystyle 2^{2}}$.

A one-dimensional space (1D space) is a mathematical space in which location can be specified with a single coordinate. An example is the number line, each point of which is described by a single real number. Any straight line or smooth curve is a one-dimensional space, regardless of the dimension of the ambient space in which the line or curve is embedded. Examples include the circle on a plane, or a parametric space curve.In physical space, a 1D subspace is called a "linear dimension" (rectilinear or curvilinear), with units of length (e.g., metre).

In algebraic geometry there are several structures that are one-dimensional spaces but are usually referred to by more specific terms. Any field ${\displaystyle K}$ is a one-dimensional vector space over itself. The projective line over ${\displaystyle K,}$ denoted ${\displaystyle \mathbf {P} ^{1}(K),}$ is a one-dimensional space. In particular, if the field is the complex numbers ${\displaystyle \mathbb {C} ,}$ then the complex projective line ${\displaystyle \mathbf {P} ^{1}(\mathbb {C} )}$ is one-dimensional with respect to ${\displaystyle \mathbb {C} }$ (but is sometimes called the Riemann sphere, as it is a model of the sphere, two-dimensional with respect to real-number coordinates).

In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

In mathematics, an algebraic number field (or simply number field) is an extension field ${\displaystyle K}$ of the field of rational numbers ${\displaystyle \mathbb {Q} }$ such that the field extension ${\displaystyle K/\mathbb {Q} }$ has finite degree (and hence is an algebraic field extension).Thus ${\displaystyle K}$ is a field that contains ${\displaystyle \mathbb {Q} }$ and has finite dimension when considered as a vector space over ${\displaystyle \mathbb {Q} }$.

The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods.

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel.

The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.

Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication.

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.

Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term homography, which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which extended Euclidean and affine spaces by the addition of new points called points at infinity. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.

In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their Cartesian coordinates, and is independent from the choice of a particular Cartesian coordinate system. The terms "dot product" and "scalar product" are often used interchangeably when a Cartesian coordinate system has been fixed once for all. The scalar product being a particular inner product, the term "inner product" is also often used.

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, the scalar product of two vectors is the product of their lengths and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the scalar product is used for defining lengths (the length of a vector is the square root of the scalar product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their scalar product by the product of their lengths).

In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as ⁠${\displaystyle A\cong B}$⁠. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified.

Geometric terms of location describe directions or positions relative to the shape of an object. These terms are used in descriptions of engineering, physics, and other sciences, as well as ordinary day-to-day discourse.

Though these terms themselves may be somewhat ambiguous, they are usually used in a context in which their meaning is clear. For example, when referring to a drive shaft it is clear what is meant by axial or radial directions. Or, in a free body diagram, one may similarly infer a sense of orientation by the forces or other vectors represented.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers ${\displaystyle \mathbb {R} }$, or a subset of ${\displaystyle \mathbb {R} }$ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of ${\displaystyle \mathbb {R} }$-vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an ${\displaystyle \mathbb {R} }$-algebra, such as the complex numbers or the quaternions. The structure ${\displaystyle \mathbb {R} }$-vector space of the codomain induces a structure of ${\displaystyle \mathbb {R} }$-vector space on the functions. If the codomain has a structure of ${\displaystyle \mathbb {R} }$-algebra, the same is true for the functions.

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.

At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.

In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.

Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.