Editing Vector space (section)

===Algebras over fields===
{{Main|Algebra over a field|Lie algebra}}
[[Image:Rectangular hyperbola.svg|class=skin-invert-image|right|thumb|250px|A [[hyperbola]], given by the equation <math>x \cdot y = 1.</math> The [[coordinate ring]] of functions on this hyperbola is given by <math>\mathbf{R}[x, y] / (x \cdot y - 1),</math> an infinite-dimensional vector space over <math>\mathbf{R}.</math>]]
General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional [[bilinear operator]] defining the multiplication of two vectors is an ''algebra over a field'' (or ''F''-algebra if the field ''F'' is specified).{{sfn|Lang|2002|loc=ch. III.1, p. 121}}

For example, the set of all [[polynomial]]s <math>p(t)</math> forms an algebra known as the [[polynomial ring]]: using that the sum of two polynomials is a polynomial, they form a vector space; they form an algebra since the product of two polynomials is again a polynomial. Rings of polynomials (in several variables) and their [[quotient ring|quotients]] form the basis of [[algebraic geometry]], because they are [[coordinate ring|rings of functions of algebraic geometric objects]].{{sfn|Eisenbud|1995|loc=ch. 1.6}}

Another crucial example are ''[[Lie algebra]]s'', which are neither commutative nor associative, but the failure to be so is limited by the constraints (<math>[x, y]</math> denotes the product of <math>x</math> and <math>y</math>):
* <math>[x, y] = - [y, x]</math> ([[anticommutativity]]), and
* <math>[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0</math> ([[Jacobi identity]]).{{sfn|Varadarajan|1974}}
Examples include the vector space of <math>n</math>-by-<math>n</math> matrices, with <math>[x, y] = x y - y x,</math> the [[commutator]] of two matrices, and <math>\mathbf{R}^3,</math> endowed with the [[cross product]].

The [[tensor algebra]] <math>\operatorname{T}(V)</math> is a formal way of adding products to any vector space <math>V</math> to obtain an algebra.{{sfn|Lang|2002|loc=ch. XVI.7}} As a vector space, it is spanned by symbols, called simple [[tensor]]s
<math display=block>\mathbf{v}_1 \otimes \mathbf{v}_2 \otimes \cdots \otimes \mathbf{v}_n,</math>
where the [[rank of a tensor|degree]] <math>n</math> varies.
The multiplication is given by concatenating such symbols, imposing the [[distributive law]] under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced in the above section on [[#Tensor product|tensor products]]. In general, there are no relations between <math>\mathbf{v}_1 \otimes \mathbf{v}_2</math> and <math>\mathbf{v}_2 \otimes \mathbf{v}_1.</math> Forcing two such elements to be equal leads to the [[symmetric algebra]], whereas forcing <math>\mathbf{v}_1 \otimes \mathbf{v}_2 = - \mathbf{v}_2 \otimes \mathbf{v}_1</math> yields the [[exterior algebra]].{{sfn|Lang|2002|loc=ch. XVI.8}}