Tensors are generalizations of vectors. These manifestate as

1. product of vector spaces
2. product of linear maps
3. 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
• 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