More on Vector Spaces and Modules

In this short post, we show a method of constructing new vector spaces (see Vector Spaces, Modules, and Linear Algebra) from old ones. But first we introduce some more definitions important to the study of vector spaces. We will refer to the elements of vector spaces as the familiar vectors. A basis of a vector space is a set of vectors v_{1}, v_{2},...,v_{n} such that any vector v in the vector space can be written uniquely as a linear combination

v=c_{1}v_{1}+c_{2}v_{2}+...+c_{n}v_{n}.

where c_{1}, c_{2},...,c_{n} are elements of the set of “scalars” of the vector space. The number of elements n of the basis is called the dimension of the vector space.

The tensor product V\otimes W is the quotient set of elements which are formal linear combinations of ordered pairs of vectors (v, w), where v\in V and v\in W under the following equivalence relations (see Modular Arithmetic and Quotient Sets)

(v_{1}, w)+(v_{2}, w)\sim (v_{1}+v_{2}, w)

(v, w_{1})+(v, w_{2})\sim (v, w_{1}+w_{2})

c(v, w)\sim (cv, w)

c(v, w)\sim (v, cw)

where c is a scalar.

The last two equivalence relations together also imply that:

(cv, w)\sim (v, cw)

Denoting the set of scalars by F, so that c\in F, we also sometimes write V\otimes_{F} W.

Tensor products also exist for modules. In physics, vector spaces provide us with the language we use to study quantum mechanics, and tensor products are important for expressing the phenomenon of quantum entanglement.

This post is quite a bit shorter than most of my previous ones and could perhaps be seen as just an addendum to Vector Spaces, Modules, and Linear Algebra. More on tensor products shall perhaps be discussed in future posts, including examples and applications.

References:

Vector Space on Wikipedia

Tensor Product in Wikipedia

Quantum Entanglement on Wikipedia

Abstract Algebra by David S. Dummit and Richard M. Foote

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5 thoughts on “More on Vector Spaces and Modules

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