### Quick description

The tensor product is a way to encapsulate the notion of bilinearity, and can be thought of as a multiplication of two vector spaces.

### Prerequisites

Linear algebra.

### See also

### General discussion

The dimension of a tensor product of two vector spaces is precisely the product of their dimensions, so when one wishes to show that a certain vector space is finite dimensional, one can try to show that it is a subspace of a tensor product (or an image of a tensor product) of two finite dimensional vector spaces.

### Example 1

Fix a field . Some notation: is the polynomial ring in one variable, is the field of rational functions, is the ring of formal power series, and is the field of formal Laurent series.

A power series is said to be **D-finite** if it satisfies a linear differential equation for some polynomials with , and . Let denote the -subspace of spanned by the derivatives of . Then the property of being D-finite can be seen to be equivalent to requiring that the subspace is finite dimensional over . From this, it is easy to see that the sum of two D-finite generating functions is also D-finite since . But what about the product of two D-finite generating functions?

We can define a map by multiplication: the pair simply goes to . The subspace spanned by the image of this will contain by the Leibniz rule for taking the derivative of a product. But this map is not linear, so we cannot say much about the dimension of this span. However, it is *bilinear*, and hence we have an associated linear map whose image is precisely the span of the image of the bilinear map, and we see then that , so is also D-finite.

### Example 2

This finite dimensionality argument is used when proving a basic result about affine algebraic groups over fields, namely that they admit a faithful linear representation (and thus are rightfully called linear algebraic groups).

An affine algebraic group is of the form where is a algebra of finite type endowed with a comultiplication . Constructing a faithful linear representation of boils down to finding a surjection where the polynomial ring is endowed with the usual Hopf algebra structure. This is done by choosing carefully a finite set of generators of the algebra in such a way that the spanned finite dimensional vector space satisfies and writing down the coefficients. See e.g Borel's book Linear Algebraic Book, sections I.1.9 and I.1.10

## Comments

## Can the notation be

Sat, 18/04/2009 - 00:45 — anonymous (not verified)Can the notation be explained? Is different from different from ?

## Clarification

Tue, 21/04/2009 - 07:25 — JoseBroxAnonymous:

Usually, all those are different algebraic structures:

represents the ring of

polynomialson one variable () with coefficients on , i.e., $K[x]=\{\sum_{i=0}^n k_i