If one considers geometrical vectors, and the operations one can perform upon these vectors such as addition of vectors, scalar multiplication, with some natural constraints such as closure of these operations, associativity of these and combinations of these operations, and so on, we arrive at a description of a mathematical structure which we call a vector space.
The "vectors" need not be geometric vectors in the normal sense, but can be any mathematical object that satisfies a set of vector space axioms. Polynomials of degree ≤n with real-valued coefficients form a vector space, for example. It is this abstract quality that makes it useful in many areas of modern mathematics.
- vector addition: defined on the Cartesian product V × V with values in V and denoted v + w, where v, w ∈ V, and
- scalar multiplication: defined on the Cartesian product F × V with values in V and denoted a v, where a ∈ F and v ∈ V.
which satisfy following axioms (for all a, b ∈ F and u, v, and w ∈ V):
- Vector addition is associative: u + (v + w) = (u + v) + w.
- Vector addition is commutative: v + w = w + v.
- There exists an additive identity element 0 in V, such that for all elements v in V, v + 0 = v.
- For all v in V, there exists an element w in V, such that v + w = 0.
- Scalar multiplication is associative: a(bv) = (ab)v.
- 1 v = v, where 1 denotes the multiplicative identity in F.
- Scalar multiplication distributes over vector addition: a(v + w) = a v + a w.
- Scalar multiplication distributes over scalar addition: (a + b)v = a v + b v.
The elements of V are called vectors and the elements of F are called scalars. In most applications the scalars are the real or complex numbers.
- A vector space over the field of real numbers R is called a real vector space.
- A vector space over the field of complex numbers C is called a complex vector space.
The formal definition of a vector space is entirely abstract, like the concept of a field itself, and analogous to the concept of a module over a ring, of which it is a specialization. To determine if a set V is a vector space, one only has to specify the set V, a field F, and define vector addition and scalar multiplication in V. Then, if V satisfies the above eight axioms, it is a vector space over the field F.
There are a number of properties that follow easily from the vector space axioms. These include:
- The zero vector 0 ∈ V (defined by axiom 3) is unique.
- a 0 = 0 for all a ∈ F.
- 0 v = 0 for all v ∈ V where 0 is the additive identity in F.
- a v = 0 if and only if either a = 0 or v = 0.
- The additive inverse of a vector v (defined by axiom 4) is unique. It is usually denoted −v. The notation v − w for v + (−w) is also standard.
- (−1)v = −v for all v ∈ V.
- (−a)v = a(−v) = −(av) for all a ∈ F and all v ∈ V.
See Examples of vector spaces for a list of standard examples.
Subspaces and bases
Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is described as being linearly independent. A linearly independent set whose span is the whole space is called a basis.
All bases for a given vector space have the same cardinality by the ultrafilter lemma (a weakened version of the axiom of choice). Using Zorn’s Lemma, it can be proved that every vector space has a basis, and vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the dimension of the vector space) representing the size of the basis. For instance, the real vector spaces are just R0, R1, R2, R3, …, R∞, …. As you would expect, the dimension of the real vector space R3 is three.
A basis makes it possible to express every vector of the space as a unique combination of the field elements. Vector spaces are usually introduced from this coordinatised viewpoint.
Given a translationally invariant and rescaling invariant topology over a vector space (preferably infinite-dimensional), the sum of an infinite sequence of vectors can be defined as the topological limit, if it exists. See topological vector space.
Main article: Linear transformation
Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products. The set of all linear maps from V to W, denoted L(V, W), is also a vector space over F. When bases for both V and W are given, linear maps can be expressed in terms of components as matrices.
The vector spaces over a fixed field F, together with the linear maps, form a category.
Generalizations and additional structures
It is common to study vector spaces with certain additional structures. This is often necessary for recovering ordinary notions from geometry. Some of these additional structures include:
- A real or complex vector space with a defined length concept, i.e., a norm, is called a normed vector space.
- A real or complex vector space with a notion of both length and angle is called an inner product space.
- A vector space with a topology compatible with the operations (i.e., such that addition and scalar multiplication are continuous maps) is called a topological vector space.
- A vector space with a bilinear operator (defining a multiplication of vectors) is an algebra over a field.
The definition of a vector space makes perfectly good sense if one replaces the field of scalars F by a general ring R. The resulting structure is called a module over R. In other words, a vector space is nothing but a module over a field.
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