Vector space

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In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. More formally, a vector space is a set on which are defined two binary operations, usually called (vector) addition and (scalar) multiplication, that satisfy certain natural axioms (listed below). Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, the sciences, and engineering.

The most familiar vector spaces are two- and three-dimensional Euclidean spaces. Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors (quantities with a magnitude and a direction, usually depicted as arrows). These vectors may be added together using the parallelogram law (vector addition) or multiplied by real numbers (scalar multiplication). The behavior of geometric vectors under these operations provides a good intuitive model for the behavior of vectors in more abstract vector spaces, which need not have a geometric interpretation. For example, the set of (real) polynomials forms a vector space.

Contents

1 Formal definition
2 Elementary properties
3 Examples
4 Subspaces and bases
5 Linear transformations
6 Generalizations and additional structures
7 References
8 See also

Formal definition

Let F be a field (such as the real numbers or complex numbers), whose elements will be called scalars. A vector space over the field F is a set V together with two operations,

vector addition: V × V → V denoted v + w, where v, w ∈ V, and
scalar multiplication: F × V → V denoted a v, where a ∈ F and v ∈ V,

satisfying the axioms below. Four require vector addition to be an abelian group, and two are distributive laws.

Vector addition is associative:
For all u, v, w ∈ V, we have u + (v + w) = (u + v) + w.
Vector addition is commutative:
For all v, w ∈ V, we have v + w = w + v.
Vector addition has an identity element:
There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Vector addition has inverse element:
For all v ∈ V, there exists an element w ∈ V, called the additive inverse of v, such that v + w = 0.
Distributivity holds for scalar multiplication over vector addition:
For all a ∈ F and v, w ∈ V, we have a (v + w) = a v + a w.
Distributivity holds for scalar multiplication over field addition:
For all a, b ∈ F and v ∈ V, we have (a + b) v = a v + b v.
Scalar multiplication is associative:
For all a, b ∈ F and v ∈ V, we have a (b v) = (ab) v.
Scalar multiplication has an identity element:
For all v ∈ V, we have 1 v = v, where 1 denotes the multiplicative identity in F.

Formally, these are the axioms for a module, so a vector space may be conscisely described as a module over a field. A vector space is thus a special case of a module.

Note that some sources may choose to also include two axioms of closure:

Vector addition is closed:
If u, v ∈ V, then u + v ∈ V.
Scalar multiplication is closed:
If a ∈ F, v ∈ V, then a v ∈ V.

However, the modern formal understanding of the operations as maps with codomain V implies these statements by definition, and thus obviates the need to list them as independent axioms.

Note that expressions of the form “v a”, where v ∈ V and a ∈ F, are, strictly speaking, not defined. Because of the commutativity of the underlying field, however, “a v” and “v a” may be treated synonymously, and this is often done in practice.

Elementary properties

There are a number of properties that follow easily from the vector space axioms.

The zero vector 0 ∈ V is unique:
If 0₁ and 0₂ are zero vectors in V, such that 0₁ + v = v and 0₂ + v = v for all v ∈ V, then 0₁ = 0₂ = 0.
Scalar multiplication with the zero vector yields the zero vector:
For all a ∈ F, we have a 0 = 0.
Scalar multiplication by zero yields the zero vector:
For all v ∈ V, we have 0 v = 0, where 0 denotes the additive identity in F.
No other scalar multiplication yields the zero vector:
We have a v = 0 if and only if a = 0 or v = 0.
The additive inverse −v of a vector v is unique:
If w₁ and w₂ are additive inverses of v ∈ V, such that v + w₁ = 0 and v + w₂ = 0, then w₁ = w₂. We call the inverse −v and define w − v ≡ w + (−v).
Scalar multiplication by negative unity yields the additive inverse of the vector:
For all v ∈ V, we have (−1) v = −v, where 1 denotes the multiplicative identity in F.
Negation commutes freely:
For all a ∈ F and v ∈ V, we have (−a) v = a (−v) = − (a v).

Examples

See Examples of vector spaces for a list of standard examples.

Subspaces and bases

Main articles: Linear subspace, Basis

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 changing the span, the set is described as being linearly independent. A linearly independent set whose span is the whole space is called a basis for V.

Using Zorn’s Lemma (which is equivalent to the axiom of choice), it can be proved that every vector space has a basis. Using the ultrafilter lemma (which is strictly weaker than the axiom of choice), one can show that all bases for a given vector space have the same cardinality. Thus 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, …. 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 tuple of the field elements. Sometimes, vector spaces are introduced from this coordinatised viewpoint.

One often considers vector spaces which also carry a compatible topology. Compatible here means that addition and scalar multiplication should be continuous operations. This requirement actually ensures that the topology gives rise to a uniform structure. When the dimension is infinite, there are generally more than one inequivalent topologies, which makes the study of topological vector spaces richer than that of general vector spaces.

Only in such a topological vector spaces can one consider infinite sums of vectors, i.e. series, through the notion of convergence. This is of importance e.g. in quantum mechanics, where physical systems are defined as Hilbert spaces, and in other areas where Fourier expansions are used.

Linear transformations

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.

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.

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.

A real or complex vector space with a well-defined concept of length, i.e., a norm, is called a normed vector space.
A normed vector space with the additional well-defined concept of angle is called an inner product space.
A vector space with a topology compatible with the operations — 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.
An affine space is a set with a transitive vector space action — informally, a vector space that has forgotten its origin.

References

Hoffman, Kenneth and Ray Kunze. Linear Algebra, Prentice Hall, ISBN 0135367972.
Seymour Lipschutz and Marc Lipson. Schaum's Outline of Linear Algebra, McGraw-Hill, 3rd edition, ISBN 0071362002.