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.
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.
a product is an operation which is distributive over addition. we call these bilinear operations. 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.
It appears that this topic is comming up alot. The tensor product is a way to combine two tensors to obtain another tensor. Suppose A and B are two vectors and A is the tensor product of the two. Then the tensor product is expressed as (Note: The symbol of the tensor product is an x surrounded by a circle. Since I don't have that symbol at my disposal I will use the symbol "@" instead.) C = A@B The meaning of this expression comes from the action of the tensor C on two 1-forms, "m" and "n". This is defined as C(m,n) = A@B(m,n) = A(m)B(n) Best wishes Pete