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Terilien

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- Thread starter Terilien
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In summary, the tensor product is a universal bilinear operation that serves as the "best" notion of multiplication for vector spaces or modules. It can be used to define other notions of a product of vectors, and it allows for a way to make bilinear maps linear. This makes it easier to handle and is a topic that is often discussed. The tensor product is expressed as the combination of two tensors, and its meaning comes from its action on two 1-forms.

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Terilien

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Hurkyl

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It is the "best" notion of multiplication for vector spaces or modules. Any other notion of a "product" of vectors can be defined by doing something to the tensor product of the vectors.

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mathwonk

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a tensor product is a universal bilinear ooperatioin such that any other biklinear operation is derived from it.

i.e. if G,H are two abelian groups, there is a bilinear map GxH-->GtensH such that f=or any other bilinear map GXH-->L, THERE IS A factorizATION OF THIS MAP via GxH-->GtensH-->L.

another point f view is that the tensor product is a way of making bilinear maps linear. i.e. the factoring map GtensH-->L above is actually linear.

linearizing things is always considered a way of making them easier to handle.

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pmb_phy

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It appears that this topic is comming up alot.Terilien said:

The tensor product is a way to combine two tensors to obtain another tensor. Suppose

The meaning of this expression comes from the action of the tensor

Best wishes

Pete

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The tensor product is a mathematical operation that combines two vector spaces to create a new, larger vector space. It is often used in multilinear algebra and has applications in fields such as physics and engineering.

The tensor product was developed to provide a way to combine vector spaces in a way that preserves their essential properties, such as dimensionality, linearity, and basis independence. This allows for a more flexible and powerful approach to mathematical and scientific problems.

The tensor product has numerous applications in physics, including in the study of electromagnetism, quantum mechanics, and general relativity. It is also used in engineering for tasks such as image and signal processing, as well as in computer science for tasks such as natural language processing and machine learning.

The tensor product differs from other vector space operations, such as addition and multiplication, in that it combines vector spaces rather than individual vectors. It also has different properties and rules, such as the ability to distribute over addition and the use of multi-index notation.

Yes, the tensor product can be extended to any number of vector spaces. This is known as the n-fold tensor product and is denoted by the symbol ⊗n. It is defined as the result of repeatedly taking the tensor product of two vector spaces n times.

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