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## Main Question or Discussion Point

could someone please explain to me what the tensor product is and why we invented it? most resources just state it without listing a motivation.

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could someone please explain to me what the tensor product is and why we invented it? most resources just state it without listing a motivation.

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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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It appears that this topic is comming up alot.could someone please explain to me what the tensor product is and why we invented it? most resources just state it without listing a motivation.

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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