The tensor quotient
The Tensor Product is an essential element in constructing rings of modules and vector spaces, as it shows that any vector space can be constructed as a product of its basis elements as defined by any of several ways to define a tensor product definitions – all leading up to an isomorphism property between objects satisfying this property, and those satisfying another definition; its meaning has even earned itself its own place within mathematics, often abbreviated as T.
A simple way of creating a tensor product is taking the product of two vector spaces. Consider two vector spaces V and W; their tensor product would then be composed of all linear maps connecting V to W; this infinite-dimensional vector space thus formed is known as its dimension(s).
Physics scientists use tensor products in various applications. For instance, they may employ them when describing the state of quantum particles which can be represented as a matrix in vector space; such matrices contain information about interactions among particles as well as probabilities.
Tensor products can also be applied in algebraic geometry. Tensor products of two fields serve as an extension of their base field and can be used to construct algebraic varieties – these varieties then serve to create various geometrical structures such as exterior algebras, symmetric algebras, Clifford algebras, Weyl algebras and universal enveloping algebras in general.
Tensor products are an integral component of algebraic geometry and an invaluable link between classical and modern mathematical theories. Furthermore, their simple nature and wide application make tensor products invaluable tools for scientists and engineers alike – not to mention being very easy to implement into computer programs; in fact, most programming languages already include built-in functions for this application!
