# Tensor

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**Tensors** are generalizations of vectors. These manifestate as

- product of vector spaces
- product of linear maps
- multi-indexed arrangements of numbers or scalar functions

In the first case we take two vector spaces , and construct a new one
by mean of their basis: if *V* has as basic vectors and *W* has
then

is the vector space generated by the symbols which are all the linear combinations of pairs .

In another hand, if we want to construct a bilinear map then we pick up a pair of covectors and then we manufacture

- defined via

It is also know that for finite dimensional vector spaces the tensor of rank one are the elements of
and its dual space

In the multi-indexed versions tensors are expressions like ,
, , which
actualy are the **components of tensors and are in nature scalars or scalar functions. In the multi-indexed**
version tensor are arrays of quantities that obeys certain rules of transformations when we change systems of coordinates.

## Comments on how to learn this

- First of all you have to be sure that you understand linear algebra
- perfectly distingishing among vector, vectorspace, vectorfunction and vectorproducts
- relations bewteen matrix, basis, linear transformation and change of coordinates (or coordinates change)

- A key concept is that
**the set of all linear transformations among vector spaces it is also a vector space** - This vector space is representable thru matrices
- Some good experience on calculus on several variables are ideal
- Tensors are objects in the branch of math called multilinear algebra which can be developed in several levels
- It is better to understand multilinear algebra over the real numbers at the beginnings
- At the same time you are studying multilinear algebra, study differential geometry at least of curves and of surfaces
- There are discrepancies among the many notations of many different authors, so compare, contrast
- You shouldn't be scare of the intense use of symbols and relations inter symbols