Vector space
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 |
2 Formal Definition 3 Examples 4 Subspaces and bases 5 Linear maps |
History
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:
- v+w belongs to V.
V is closed 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 -v in V, such that v+(-v)=0.
Existence of additive inverses in V. - v+w=w+v.
Commutativity of vector addition in V. - a*v belongs to V.
V is closed 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.
From the above properties, one can immediately prove the following handy formulas:
- a*0 = 0*v = 0
- -(a*v) = (-a)*v = a*(-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.
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: The vector space R^{n}, over R, with component-wise operations
- More generally, F^{n}, 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(p^{n}), 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 R^{0}, R^{1}, R^{2}, R^{3}, ..., R^{∞}, ... As you would expect, the dimension of the real vector space R^{3} 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.