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The fundamental concept in linear algebra is that of a vector space or linear space.
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 actually geometric vectors, but can be any mathematical object that satisfies the following 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.
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 length concept, i.e., a norm, is called a normed vector space.
Examples
See Examples of vector spaces.
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 modules over R. In other words, a vector space is nothing but a module over a field.
See also
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