The following sections will develop the primary notion of multilinear algebra from the minimal requirements of you were being exposed to a course of linear algebra (and if vector calculus is already in your experiences, better). This sections are written in the language of Shakespeare cuz (because) science is better written-in or transmited-on (and more efficiently) than in spanish.
Latter, you should be able to understand a many examples of applications: firstly on geometry and in physics simultaneously or subsequently. Please visit these notes frequently because they are going to be upgraded continuosly, and… enjoy!
A essential key to understand tensors is to use the linear algebra and the several-variables-calculus that you already should know.
Do you remember that linear transformations can be added among them and multiplied by scalars? then you could understand that the dual space , of the vector space , is another vector space: if we set
and if are two arrows then define another element of . It also makes sense if then defines another linear functional.
Exercise: write down the definitions of and and give three concrete examples. For the record: the elements of are also called covectors.
An example of a covector defining a linear transformation is via a pairing
i.e. multiplication of matrices:
A multilinear scalar is a function defined over products of vector spaces to the scalar base field which is linear on each slot of the arguments. In other words, considering each factor of
For example: is bilinear if
A general method to construct multilinear scalars is by means of the tensor product of covectors: if , then their tensor product is
it is easy to check that this construction is a bilinear application. Also generalizable in the following sense: if and are two covector in different vector spaces still the formula produce a bilinear assignation. Augmenting factors is not a problem. A trilinear transformation is a map which is linear in each slot and it can be constructed via
where , and .
Question: if is a trilinear map: how is it that is related with the triple tensor product above?
Remark: since the sum of linear transformations gives other linear tranformation we could construct the vectorspace of linear transformations and in the same manner it is posible to give to the set of all bilinear maps from into the scalar field, namely
the estructure of vectorspace: summing two bilinear maps is not a problem as far as acting scalar over them. Ahead we are gonna learn how to distiguish base for all these machines.
The tensor product of vector spaces is a device to construct new vector spaces form old ones. Let us explain with an example: suppose that we want to build a vector space from to initial ones and , vector spaces over the real numbers, and both are finite dimensional, say and . So is generated by basic vectors . It is obvious that if then for some scalars , and similarly for with as base vectors. So let us construct the tensor product of and , symbolized
It is the vector space whose base vectors are the symbols . This means that if then this vector must be a linear combination
here the scalars are called components of tensor . Clearly .
The Einstein sum convention allow us to write for vectors in but
Basis for covector and tensor products. One can check inmediatly that given a vectorspace and a base for them, in symbols:
that the defined assignations determine linear maps via linear extension:
and where we exactly know what it means . Hence we can appreciate that the are simple projectors. With them we will be in position of describe a basis in , in and more.
Simply the vector space is is generated by . If then and then
Let us describe the basis of . Knowing that then for any and if is another covector then
Observe that if are two base vectors then
How do the components of a tensor vary when the base vectors are changed? A change of basis in is determined by
or simply . So the matrix of the change of basis is : i-rows and j-columns.
Let us remember how the components of a vector change when we have a :
if where then
How do the components of a matrix of a linear transformation change, when we change basis and ?
Tensors in components
Given a pair of order 2 tensor and entonces
In other words the components of
Or if then
Wedge Product of Covectors
Given two covectors, , we define their wedge product as
With this we have a bilinear alternating-map:
evaluated at the basis
And what is the triple wedge product?
where are basic three covectors.
lección uno en pdf, versión preliminar con misprints y errores a proposito o adrede. Reporta-los para obtener puntos en la calificación final. Pulsa: https://juanmarqz.files.wordpress.com/2009/10/leccionuno.pdf