## Tensor as an element of tensor product of vector spaces

Before presenting another way of definition a tensor, we define a notation. A linear map and a bilinear form are respectively written as a linear combination of and . Any of these (for any the dimensions of the corrresponding spaces) can be considered as one new object and denoted as for example. and . The writing of the basis vectors and/or basis covectors adjacent to each other is usually denoted by and . This notation is referred to as tensor product of (basis) vectors. A general definition will be presented later. Using this notation, for now, we can write a linear map and a bilinear form as,

This notation can be extended to be used with any finite linear combination of tensor products of basis vectors and/or covectors where the combination coefficients takes indices following the index level convension. For example we can write,

Let’s define tensor product of vectors and covectors and their rules.

### Tensor product of vectors and covectors

Let , and be vectors or covectors (not necessarily basis ones), the we define the tensor product of each pair as , and etc., and the following rules and operations,

0. Order matters:

- Scalar multiplication: .
- Addition: and .

The above rules can be extended to tensor product of any number of vectors or covectors. For example,

1. Scalar multiplication:

2. Addition:

The above rules can be recruited to construct vector spaces, called tensor-product vector spaces. For example, if and , then,

Any vector spaces can get into a tensor product. For example, with members like with and .

Note that tensor product of vector space can be done on totally different vector spaces over the same field, e.g. .

**Basis for a tensor product space**

Let and be vector spaces with bases and \{\zeta_j\}_1^m respectively. if and , we can write,

This states that any vector can be written as a linear combination of . Therefore,

- The set of vectors \} is a basis for the vector space .
- The dimension of the vector space is .

The above can be extended to tensor product of any number of vector spaces, i.e. the tensor product of the basis vectors of vector spaces creates a basis for the resultant tensor product space.

**Example:** Let and and .

Then, is a basis for , and,

### Tensor by tensor product

**Definition (Tensor-product view):** Tensor is a collection of vectors and covectors combined together by using the tensor product (of vectors and/or covectors). A tensor of type (*r,s*) is a member of the tensor product space,

and written as,

Note that collects the component or the coordinates of the tensor with respect to the basis vectors \{\varepsilon^{i_1}\otimes\cdots\otimes \varepsilon^{i_r}\otimes e_{j_1}\otimes\cdots\otimes e_{j_s}\} or inherently

In this view, a vector is a (0,1) tensor, a covector is a (1,0) tensor. A linear map is a (1,1) tensor. A bilinear form is a (2,0) tensor. A bilinear map is a (2,1) tensor.