Vector space

The fundamental concept in linear algebra is that of a vector space or linear space. This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics.

Table of contents

1 Formal definition

1.1 Terminology

2 Examples
3 Subspaces and bases
4 Linear maps
5 Generalization
6 Vectors in physics
7 History

Formal definition

A set V is a vector space over a field F (for example, the field of real or of complex numbers) if, given

an operation vector addition defined in V, denoted v + w (where v, w ∈ V), and
an operation scalar multiplication in V, denoted a * v (where v ∈ V and a ∈ F),

the following ten properties hold for all a, b ∈ F and u, v, and w ∈ V:

v + w belongs to V.
(Closure of V under vector addition.)
u + (v + w) = (u + v) + w.
(Associativity of vector addition in V.)
There exists a neutral element 0 in V, such that for all elements v in V, v + 0 = v.
(Existence of an additive identity element in V.)
For all v in V, there exists an element w in V, such that v + w = 0.
(Existence of additive inverses in V.)
v + w = w + v.
(Commutativity of vector addition in V.)
a * v belongs to V.
(Closure of V under scalar multiplication.)
a * (b * v) = (ab) * v.
(Associativity of scalar multiplication in V.)
If 1 denotes the multiplicative identity of the field F, then 1 * v = v.
(Neutrality of one.)
a * (v + w) = a * v + a * w.
(Distributivity with respect to vector addition.)
(a + b) * v = a * v + b * v.
(Distributivity with respect to field addition.)

Properties 1 through 5 indicate that V is an abelian group under vector addition. The rest, properties 6 through 10, apply to scalar multiplication of a vector v ∈ V by a scalar a ∈ F. Note that property 5 actually follows from the other 9.

From the above properties, one can immediately prove that, for all a ∈ F and v ∈ V,

a * 0 = 0 * v = 0

(−a) * v = a * (−v) = −(a * v).

It can be shown that the additive inverse to every element v in V is unique. Hence we can define a function called "−" (minus) such that

v + −(v) = 0.

Furthermore, it can be proven that − o − = I, where o denotes function composition and I is the identity function. In other words, for all v,

−(−(v)) = v.

The concept of a vector space is entirely abstract, like the concepts of a group, ring, and field. 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 ten properties, it is a vector space over the field F.

The members of a vector space are called vectors.

Terminology

A vector space over R, the set of real numbers, is called a real vector space.
A vector space over C, the set of complex numbers, is called a complex vector space.
A vector space with a defined distance concept, i.e., a norm, is called a normed vector space.

Examples

Example 1: For all n Rⁿ forms a vector space over R, with component-wise operations.
- More generally, for an arbitrary field F, Fⁿ forms a vector space over F, with component-wise operations.
Example 2: The set of m × n matrices with complex elements forms a vector space over C.
- More generally, the set of m × n matrices with elements in an arbitrary field F form a vector space over F.
Example 3: The set of all continuous real-valued functions on a closed interval.
Given a vector space V over F and some set X, the set of all functions ƒ : X → V forms a vector space over F
The set of all polynomials with coefficents out of F, denoted F[x], forms a vector space over F.
The finite field GF(pⁿ) forms a vector space over GF(p).
C forms a vector space over R.
R forms a vector space over Q.

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. 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.

Linear maps

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.

An isomorphism is a linear map that is one-to-one and onto. If there exists an isomorphism between V and W, we call the two spaces isomorphic; they are then essentially identical.

The vector spaces over a fixed field F, together with the linear maps, form a category.

Generalization

Instead of using a field F for the scalars, one can also use a general ring R. Then one obtains moduless over R. In other words, a vector space is nothing but a module over a field.

Vectors in physics

Generally put, vectors in physics are “arrows” (geometrically, not categorically) which obey the mathematical definition above. The most basic physical vector is the displacement vector from point A to point B (its direction is from A to B and its length is the distance between A and B).

The other important property of a physical vector is its behavior under changes of coordinate system. See tensor for a more detailed discussion of this.

An orthogonal transformation U is a linear transformation, or, in other words, a matrix) which satisfies U · U^T = I,

where U^T denotes the transpose of T and I is the identity matrix).

It immediately follows that the determinant of an orthogonal matrix is det U = ± 1.

Transformations with det = 1 are called proper rotations, while transformation with det = −1 are called improper rotations. Intuitively, improper rotations also perform an inversion of the axis, and are therefore sometimes called “mirror operations.”

Polar vectors—such as the displacement, velocity, electric field, or linear momentum—go through transformation in the following manner:

a′ = U · a.

Axial vectors—such as the angular velocity, magnetic field, or angular momentum—go through transformation in the following manner:

b′ = (det U) · U · b.

Most of the axial vectors are related to the vector product.

History

See Linear algebra.