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

Table of contents
1 History
2 Formal Definition
3 Examples
4 Subspaces and bases
5 Linear maps


See Linear algebra.

Formal Definition

Simply put, a vector space over a field F is just an F-module.

Expanding, it means the following:

A set V is a vector space over a field F (such as the field of real or of complex numbers, for example), if given an operation vector addition defined in V, denoted v+w for all v, w in V, and an operation scalar multiplication in V, denoted a*v for all v in V and a in F, the following 10 properties hold for all a, b in F and u, v, and w in V:

  1. v+w belongs to V.
    V is closed under vector addition.
  2. u+(v+w)= (u+v)+w.
    Associativity of vector addition in V.
  3. 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.
  4. For all v in V, there exists an element -v in V, such that v+(-v)=0.
    Existence of additive inverses in V.
  5. v+w=w+v.
    Commutativity of vector addition in V.
  6. a*v belongs to V.
    V is closed under scalar multiplication.
  7. a*(b*v)=(ab)*v.
    Associativity of scalar multiplication in V.
  8. If 1 denotes the multiplicative identity of the field F, then 1*v=v.
    Neutrality of one.
  9. a*(v+w)=a*v+a*w.
    Distributivity with respect to vector addition.
  10. (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. Properties 6 through 10 apply to scalar multiplication of a vector v in V by a scalar a in F. (Note that Property 5 actually follows from the other 9.)

From the above properties, one can immediately prove the following handy formulas:

a*0 = 0*v = 0
-(a*v) = (-a)*v = a*(-v)
for all a in F and v in V.

The members of a vector space are called vectors. 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 must specify the set V, a field F and define vector addition and scalar multiplication in V. Then if V satisfies the above 10 properties it is a vector space over the field F.



  • Example 1: The vector space Rn, over R, with component-wise operations
    • More generally, Fn, over F, with component-wise operations
  • Example 2: The set of (mxn) matrices with complex elements over C
    • More generally, the set of (mxn) matrices over an arbitrary field 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, then the set of all functions X -> V forms a vector space over F
  • F[x]: The set of all ploynomials with coefficents out of F, over F.
  • The finite field GF(pn), over GF(p)
  • C, over R
  • R, over Q (the rational numbers)

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 called 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, 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 is denoted L(V,W) and makes up a vector space over the same field. 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.