When and why did Mathematicians saw a need to define Tensor Products?
I want to know the historical development of the idea "Tensor Product"?
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The tensor product is actually a very simple concept.
It goes back to Babylonian times when people realised that two edges describes an area. Intuitively they realised that geometric area was bilinear but all this wasn't formalised until the twentieth century.
Mathematicians, being mathematicians, generalised by allowing the edges to take values in any vector space, as well as the attendant geometric area.
That formalisation took so long shouldn't be such a surprise. After arithmetic was invented well before Babylonian times and yet was only formalised with Peano's axioms in the early 20th C. I'd also add that it has taken the subject so far away from its geometric roots that its difficult to see the geometry of a tensor.
Its worth adding that the tensors of general relativity, which made the subject famous, are actually fields of tensors, that is tensor fields. Moreover, the tensors they use are of a special kind, comprising of a tensor power of a vector space tensored with a tensor power of its dual space.