# Thread: Cross product vs Tensor product

1. ## Cross product vs Tensor product

A certain someone here doesn't seem to know basic notation so I decided to clear it up.

Cross/Vector product
Cross product, also known as vector product, is a binary operation, typically in , which turns in an algebra over the field . Of course there are any number of possible way to make into an algebra over the field but the main point with the cross product is that it satisfies a number of desirable properties for general usage. An important property it has is that it is not associative. That is

and what is important here to notice is that we use the for cross product. Why do we do that? Well before more advanced mathematics came about to understand what things really were there were only two kinds of products for vectors, the scalar/dot product and the vector/cross product and as the name suggests, they used the two methods to write an ordinary product. That is how it has been done throughout all ages and is still to this day. In abstract algebra we use ordinary product symbols to symbolize any possible form of product that we desire, regardless of what it actually does. However it is important to know that the tensor product originates from quaternions/hamiltonians where it was just thought of as a product, just like with complex or real numbers and as such, writing an ordinary product number makes even more sense. Both vector products can be used to describe quaternions.

Tensor product
Tensor product is a relatively new development, It is a method to turn bilinear functions into linear ones by creating a new space/module that is considerbly larger than the previous ones. This one cannot be used in ordinary multiplication and has historically never been associated with anyform of "ordinary" product as percieved. that is why it got an entirely new symbil, namely , it is however, as said, NOT a product of things, that is it is not a binary operation such that

or the likes, what it is is a method to construct a new vector space, as such you cannot take elements and apply it to them to get a new element, when we write we do NOT mean a new element in our original sapce/module, we mean an ENTIRELY NEW element in a DIFFERENT space/module that can be described by using elements of the previous spaces/modules. As such it is important to keep those notations apart.

Summa summarum
For a given vectorspace we have that is a binary operation, but is not, it is a functor that produces a new vectorspace/module.

2. ## Re: Cross product vs Tensor product

The tensor product maps the vector space into which is a different vector space, so the notation is misleading; it should be (upper or lower indices depending on co- or contra variance, if such notation is relevant at all.

Whether the original vector space remains (the new space is "created" in addition) or whether the original vectors space is "destroyed" is a question of notation and context (one can destroy the "cross product" by simply changing the order or multiplication and adding) in a casual presentation.

However, arithmetic processes between scalars in a given vector space do not affect the sign of the vector product. It is the vector itself that changes sign (a "mirror" rotation in Lorentz space). (That is why the strict definition of cross product is really:

and

Changing the sign of scalars is a different process than changing the parity of the vectors.

The "outer" product is distinguished for the cross product (which relates to n=2 and the "right hand" rule) for this reason (and a number of others, including parity).

The problem the cross product raises is apparent in the problem of gimbel locking, which is why quaternions are used in action adventure games requiring rotation, among other applications.

3. ## Re: Cross product vs Tensor product

Originally Posted by BuleriaChk
The tensor product maps the vector space into which is a different vector space, so the notation is misleading; it should be (upper or lower indices depending on co- or contra variance, if such notation is relevant at all.

Whether the original vector space remains (the new space is "created" in addition) or whether the original vectors space is "destroyed" is a question of notation and context (one can destroy the "cross product" by simply changing the order or multiplication and adding).

The "outer" product is distinguished for the cross product (which relates to n=2 and the "right hand" rule) for this reason (and a number of others, including parity).

The problem the cross product raises is apparent in the problem of gimbel locking, which is why quaternions are used in action adventure games requiring rotation, among other applications.
If you read what I said I said that tensor product is a functor, a binary functor that takes 2 modules and produces a new one. Add onto that one that has a very specific universal property.

What "Wetherthe original vector space remains or wether the original vector space is destroyed" is non-sense. You cannot destroy or do anything to such. We have them always and produce new ones with various properties we desire. Tensor product are spaces that have a very specific universal property tied to them. I bet you cannot name it.

As for gimbal lock, it is irrelevant in mathematics as we are dealing with just specific logical constructions. Cross product is just a binary operation for a vector space that turns it into an algebra, that is all it is in abstract mathematics and it has no greater importans than any other. You still think like a dumbarse engineer.

4. ## Re: Cross product vs Tensor product

My proof of Fermat's Theorem is based on the realization that number theorists can only think in one dimension.
(So they have trouble defining division rings)

Edit: I didn't say the line I deleted correctly; back in a moment.
Update: back again...

In the equation , , is aninteger because it is divisible by by (a + b). For , (an integer) cannot be divided by (a+b) and remain an integer unless (a + b) = d (an integer), in which case c = d.

That is why a fraction requires two dimensions in the equation z = y/x, and why y=0 means that z is undefined. Since 0 is the midpoint of all possible metrics, if the metric for the dimension represented by x does not exist (there are no lengths) then the operation of division is undefined.

This problem arises from the constructivist perspective of number theory which proposes that numbers are created by the set of axioms initiated by Frege, and expanded to groups, rings, and fields (assumed to work in the same dimension). However, a fraction is the slope of a line in a two dimensional Cartesian space This is the foundation of STR if one replaces x by r, y by t, and z by v.

The result is then where the velocity does not change sign, but the vector r does if the r.h.s. order of multiplication is reversed. (That is why Pauli ignores the "missing" matrix).

This is the foundation of the Pauli and Dirac characterization of "number interaction" if the construction of numbers is a Lorentz rotation in (ct,vt') space (i.e. () space ) from the Lorentz transform relating the vectors (x,t) of time and space in its original interpretation, but where I am just using x and t to represent real numbers in a two dimensional Cartesian space.

This perspective begins from independence of coordinates and assumes continuity for two metrics (i.e., the relativistic unit circle), so is different from the constructivist approach (which begins with counting integers in one dimension (i.e., a line)).

For me, that is why Wiles' proof in terms of "modules" (I believe the Wiki article mentions them, as well as "curvature", but I'll check again) is suspect to me, even though I have not read the proof.... and if there is a proof, I suspect it is equivalent to the vector formulation I did....

5. ## Re: Cross product vs Tensor product

Originally Posted by BuleriaChk
My proof of Fermat's Theorem is based on the realization that number theorists can only think in one dimension.
They don't, you are nothing compared to them. We mathematicians do not think of dimensions are someone as lowly as you do. We are vastely superior to you.

Originally Posted by BuleriaChk
That is why a fraction requires two dimensions in the equation z = y/x, and why y=0 means that z is undefined. Since 0 is the midpoint of all possible metrics, if the metric for the dimension represented by x does not exist (there are no lengths) then the operation of division is undefined.

This problem arises from the constructivist perspective of number theory which proposes that numbers are created by the set of axioms initiated by Frege, and expanded to groups, rings, and fields (assumed to work in the same dimension). However, a fraction is the slope of a line in a two dimensional Cartesian space .
First of all you should know that the concept of "dimension" is not always applicable in any of thsoe things depending on circumstances so do not think that it is that important.

Second, a fraction is not "a slope" as you say it, a fraction, from a ring, is just an expansion of the ring which is vastely more diverse than your puny brain can imagine. Look up localization of rings.

The rest is irrelevant as we talk mathematics, not fucking physics.

6. ## Re: Cross product vs Tensor product

Originally Posted by emperorzelos
They don't, you are nothing compared to them. We mathematicians do not think of dimensions are someone as lowly as you do. We are vastely superior to you.

First of all you should know that the concept of "dimension" is not always applicable in any of thsoe things depending on circumstances so do not think that it is that important.

Second, a fraction is not "a slope" as you say it, a fraction, from a ring, is just an expansion of the ring which is vastely more diverse than your puny brain can imagine. Look up localization of rings.

The rest is irrelevant as we talk mathematics, not fucking physics.
Gee, what an intelligent, well posed response. Of course one would like to define an "expansion" of the ring...
Superior? I think not.......

Number theorists: "If I ignore Descartes, maybe he'll go away.. (and, after all, neither Descartes nor Newton were "mathematicians").

7. ## Re: Cross product vs Tensor product

Originally Posted by BuleriaChk
Gee, what an intelligent, well posed response. Of course one would like to define an "expansion" of the ring...
Superior? I think not.......

Number theorists: "If I ignore Descartes, maybe he'll go away...
There are many ways to expand rings, localization is one of them so get over it you imbecile. We are superior to your limited understanding that cannot even understand the differens between tensor and cross product, and let's not forget the fact that you do not even know what transcendental means.

8. ## Re: Cross product vs Tensor product

Originally Posted by emperorzelos
There are many ways to expand rings, localization is one of them so get over it you imbecile. We are superior to your limited understanding that cannot even understand the differens between tensor and cross product, and let's not forget the fact that you do not even know what transcendental means.
So you've replaced one ill defined concept ("expansion" in this thread) with yet another - "localization".

Which calls into serious question whether you understand the difference between a scalar and a vector...

Does "localization" mean the null vector by any chance? (i.e., the connection of vectors multiplied by scalars = 0 at an origin?) Just askin' Or does it mean ? Or maybe ?

(a cross ('outer") product for n > 2 is a relation (transformation) between two independent vectors to another in a third (dependent) vector space; a tensor product is a relation between vector transformations within the same vector space, so is more general (i.e., a transformation of a transformation where the determinant is a multi-linear function of indices on the transformation (i.e., the structure of the transformation is characterized by permutation groups on the indices)).

In your fantasies... dream on - and may they continue to be nightmares.... in another dimension

9. ## Re: Cross product vs Tensor product

Originally Posted by BuleriaChk
So you've replaced one ill defined concept ("expansion" in this thread) with yet another - "localization".

Which calls into serious question whether you understand the difference between a scalar and a vector...

Does "localization" mean the null vector by any chance? (i.e., the connection of vectors multiplied by scalars = 0 at an origin?) Just askin' Or does it mean ? Or maybe ?

(a cross product is a relation between two independent vectors two another in a third (dependent) vector space; a tensor product is a relation between vector transformations, so is more general (i.e., a transformation of a transformation where the determinant is a multi-linear function of indices on the transformation (i.e., the structure of the transformation is characterized by permutation groups on the indices)).

In your fantasies... dream on - and may they continue to be nightmares.... in another dimension
Look up my previous post, I included localization already in it you dimwit.

A scalar and a vector? I am sure you couldn't even give the definition of thigns at the deepest level of anything. So far you have not.

Localization has nothing to do with vectors necciserily.

And as per usual you got no clue about mathematics, grow up already and face the fact I am superior to you in mathematics

10. ## Re: Cross product vs Tensor product

And I thought John Gabriel had a problem with slope (not to mention curvature). At least he (almost) recognized two dimensions.... (well, ok, sort of, anyway)...

But I certainly see where the issue of disagreement lies... whether more than one dimension is admissible in the context of a mathematical discussion.... (so there is a religious faith that Cartesian coordinates are not somehow "real mathematics", and yet one can mention the (Cartesian) mapping without having to worry about ...

Are you SURE you are not John Gabriel... ? You sure sound like him

Page 1 of 4 123 ... Last

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•