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dimension10
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I've searched everywhere about tensor products but I just can't understand them. Can anyone please explain this concept to me?
Bacle said:If you can tell us what you don't understand, we may help you better.
If you are talking about vector spaces, then the tensor product V(x)W
gives you a new vector space in which every bilinear map from VxW into
a third space Z becomes a linear map from V(X)W--->Z .
The existence of the tensor product follows from some algebraic lemmas
that guarantee that certain maps factor through; conditions on the kernel
of homomorphisms that allow a bilinear map VxW-->Z to factor through
V(X)W.
But if we don't know your background, or more specifically where you are
stuck, it is difficult to suggest something.
quasar987 said:Tensor products are about linear and bilinear maps between vector spaces (in the simplest case!). And they are substantially more difficult to grasp than those. So I suggest you start by understanding linear and bilinear maps on vector space.
Bacle said:dimension10:
Read your definitions more carefully. A map can be linear or bilinear, but
not so for a vector space.
I don't know if you are thinking of tensoring linear maps, maybe, but
even then, you are kinda off.
mathwonk said:a dot product is a bilinear map. a tensor product is a technical device which linearizes all bilinear maps.
see my notes on my web page, or search my many posts here for this topic.
A tensor product is a mathematical operation that combines two vector spaces to create a new, larger vector space. It is used to represent the relationship between two different vector spaces and can be thought of as a generalization of the outer product between two vectors.
The main purpose of using tensor products is to represent and mathematically manipulate multidimensional data. It allows for the analysis of complex systems by breaking them down into simpler components and examining their relationships.
Unlike a regular product, which results in a single value, a tensor product results in a multidimensional object. The dimensions of the tensor product are determined by the dimensions of the two vector spaces being multiplied together.
The calculation of a tensor product involves taking the outer product of two vectors and then using a tensor product rule to combine the resulting matrices. This can be done using a tensor product notation or by using matrix operations.
Tensor products have various applications in fields such as physics, engineering, and computer science. They are used in quantum mechanics, signal processing, image processing, and machine learning, to name a few. Tensor products are also essential in the study of vector spaces, linear transformations, and multilinear algebra.