Vector spaces are used to model a collection of objects – vectors – that can be combined in a linear way to form more vectors. Vector spaces may also be called linear spaces.
Formally, a vector space over a field (with usually being either or ) of "scalars" is a set together with two operations:
Vector Addition. The sum of two vectors and is written as . Its properties are as follows for all vectors , and :
Scalar multiplication. Combining the scalar with the vector results in a vector . Scalar multiplication satisfies the following properties for all scalars and vectors :
A vector subspace is a subset of some vector space that is itself a vector space when inheriting vector addition and scalar multiplication from . Equivalently, is a vector space whenever it is closed under vector addition and scalar multiplication.
A linear combination of vectors is an expression of the form for some sequence of scalars . These vectors are said to be linearly independent if their linear combination is zero if and only if .
The span of a finite set of vectors is the set of all linear combinations of those vectors. Spans are themselves vector spaces.
For a given vector space , a basis of the space is a sequence of linearly independent vectors whose span is equal to . If such a basis exists (and the afformentioned sequence is finite), then is said to be finite dimensional with a dimension equal to the number of vectors making the basis. The dimensionality of a vector space, when defined this way, is independent of the choice of basis.
A vector space is said to be infinite dimensional if it is not finite dimensional.
Some common examples of vector spaces include:
The (algebraic) dual space of the vector space over the scalar field consists of all linear mappings of the form . is itself a vector space over . Elements of are referred to as linear functionals or covectors.
If is finite-dimensional, then shares the same dimension. That is, and are isomorphic. Moreover, for any non-degenerate bilinear form , the bijection is an isomoprhism between and . Thus, a finite-dimensional inner product space has a "natural" isomorphism to its dual.
Regardless of the dimensionality of , there is a natural homomorphism from to is dual of its dual, . This is because the "evaluation map" corresponding to , defined by , is clearly am element of o . And the relationship to of a vector and its evaluation map are clearly homomorphic. Moreover, this homomorphism is an isomorphism for finite-dimensional . Because this isomorphism is natural, double duals and other "higher order duals" are usually ignored.
A homomorphism is a mapping that preserves algebraic structure. In the context of group theory, a homomoprhism between a group and is a mapping with the following property for all :
may be categorized as a special "type" of homomorphism according to additional properties it may hold:
A topological space is a set equipped topological structure. Essentially, a topological space is defined so that the "limit" of a sequences o "points" in the space may be defined.
A topological space may be defined as a set to together with a topology , which consists of subsets of . Elements of this topology are called open sets. A topology satisfies the following axioms:
A set is said to be closed if its complement is open. The set of closed sets uniquely specifies space's topology.
For a given point , the neighborhood with respect to a topology on as the collection of open sets containing . The collection of neighborhoods uniquely specifies a space's topological structure.
Using neighborhoods, it is possible to define topological spaces without first referring to topologies. Consider a mapping of the form satisfying the following properties for all :
If these properties are specified, then maps to its neighborhood with respect to some topology. That is, the above enumerated properties are an alternative and equivalent axiomatization of a topological space.
Useful topological spaces are usually not defined by explictly specifying the topology directly. Instead, topological structures are typically generated from simpler constructions:
A bilinear form over a vector space (over some scalar field ) is a mapping that is linear in each argument.
Bilinear forms may be characterized by additional properties they posses. Let be a bilinear form. Then is:
non-degenerate if for all only when ,
symmetric if for all ,
skew-symmetric if for all ,
alternating if for all (alternating implies skew-symmetric).
reflexive if implies that .
Bilinear forms may be represented as matrices for finite-dimensional vector spaces. Let . And every bilinear form can be defined in the following fashion:
where is the matrix representation of , and and are treated as column vectors.
An group is a simple algebraic structure that represents a composable set of permutations. There are two popular definitions of a group: one as a set of permutations and one as a set equipped with a binary operator. Both definitions are essentially equivalent by Caley's Theorem.
One formulation of a group is as a set of permutations. That is, a permutation group is a set of bijections of some set with the following closure properties:
An abstract group is a set together with a mapping called a binary operator. We write as or simply as .
An abstract group must satisfy the following properties:
It is clear that a permutation group is an abstract group by having composition as the chosen binary operator. Caley's theorem states that the reverse is true in a sense.
Note: For notational convenience, the binary operator is usually assumed and a group is identified by the underlying set. So, for example, claiming that " is an element of the group " means that " is an element of the underlying set of the group ".
If , then is a subgroup of if is itself a group. Equivalently, whenever is is closed under inversion and composition.
A group is abelian if its binary operator is commutative. That is, whenever for all .
An inner product space is a vector space over a scalar field together with a bilinear form , called an inner product, that satisfies the following properties:
The function defined by satisfies the properties of a norm (thus an inner product space is a nomed vector space). The inner product can be recovered from its norm using the polarization identity:
The inner product norm further satisfies the famous Cauchy-Schwartz inequality:
Caley's theorem asserts that a permutation group and an abstract group are essentially equivalent concepts. It is clear that a permutation group meets the definition of an abstract group. Caley's theorem states that an abstract group is isomorphic to some permutation group.
For a given , the bijection defined by is the left action by . Similarly, the map is the right action by . The mapping from to the corresponding left action (or right action) is an isomorphism.
Galilean Relativity refers to a theory of relativity consistent with Newton's laws.
It is named after Galileo's thought experiment involving a vessel traveling with a perfectly uniform and linear motion. An observer contained within the vessel, observing only phenomena also contained within the vessel, will have no means of determining the vessel's speed or direction of travel.
A galilean space(time) is a four-dimensional affine space . Points in this space are called "events". The set of displacements forms a four-dimensional, real vector space .
There exists a rank-1 linear map mapping spatio-temporal displacements to time intervals. Two events and are simultaneous if .
The three-dimensional quotient space is euclidean (that is, equipped an inner product ). From this, the distance between simultaneous events and is defined as
where is the natural projection (epimorphism) from to .
An isomorphism between galilean spaces is a bijection that preserves galilean structure (affinity, euclidean distance between simultaneous events, and time intervals).
All galilean spaces are isomorphic to , where the and components are each equiped with the standard inner product. Isomorphisms of spacetime onto are called inertial reference frames or galilean coordinates.
Automorphisms of are called galilean transformations, which forms the galilean group. The galilean group is generated from the following galilean transformations:
Let be a group and be a vector space over the field . A representation with respect to these objects is a homomorphism from to , the set of isomophisms of . That is, a representation is just a group action consisting of linear transformations. The study of group representations forms much of representation theory.
Usually, and is finite. In this case, (or, rather, ) may be identified as a finite set of matrices under some coordinate system.
The subspace is said to be invariant with respect to if for all and .
The representation is said to be irreducible over if the only invariant subspaces are and the subspace consisting of just the zero element.
An important problem of representation theory is decomposing into irreducible components That is, write with irreducible and none of is equal to . If this is possible, then the representation is said to be fully reducible.
A normed vector space is a vector space over together with a function , the norm, satisfying the following properties for all and :
A normed vector space is also a metric space under the metric .
A seminorm has the properties of the norm, except with the property that may be zero for nonzero .
The dual numbers can be thought of an extension of the real numbers with an infinitesimal offset. A dual number may be written as , where and are real numbers. This expression is linear in and and is said to obey the following rule of multiplication:
In particular, .
For a polynomial , it can be readily shown that:
And for any analytic function , one can similarly extend to the dual numbers:
With this, it is possible to "automatically differentiate" a function by calculating the "epsilon" component of . The practicality of this approach depends on how is written.
The ForwardDiff.jl Julia package follows this exact approach, utilizing a multidimensional analogue of dual numbers to calculate gradients.
A quotient group for a given group and a "normal" subgroup of is a group that has a "coarser" or "more relaxed" algebraic structure relative to . The notion of a quotient group is essential in a number of isomorphism theorems.
Let be a group with subgroup .
A left coset of , denoted as or for some , is the set consisting of elements of the form for some . That is, is the image of under the left action induced by .
Similarly, a right coset of , is the image of under the right action induced by .
The subgroup is said to be normal if for all . In this case, there is no distinction between "right" and "left" cosets.
The cosets of a normal subgroup itself forms a group
Let and be finite and normal. Lagrange's theorem states that