I think the people who find this definition to be mysterious are really looking for (borrowing from Ravi Vakil[0]) "why is a tensor" rather than "what is a tens

I think the people who find this definition to be mysterious are really looking for (borrowing from Ravi Vakil[0]) "why is a tensor" rather than "what is a tensor". In that case, a better answer IMO is that it's the "most generic" way to multiply vectors that's compatible with the linear structure: "v times w" is defined to be the symbol "v⊗w". There is no meaning to that symbol. But these things are vectors, so you could write e.g. v = a⋅x+b⋅y, and then you want e.g. (a⋅x+b⋅y)⊗w = ax⊗w + by⊗w, and so on. So in some sense, the quotient space construction[1] gives a better "why". It says * I want to multiply vectors in V and W. So let's just start by writing down that "v times w" is the symbol "v⊗w", and I want to have a vector space, so take the vector space generated by all of these symbols. * But I also want that (v_1+v_2)⊗w = v_1⊗w + v_2⊗w * And I also want that v⊗(w_1+w_2) = v⊗w_1 + v⊗w_2 * And I also want that (sv)⊗w = s(v⊗w) = v⊗(sw) And that's it. However you want to concretely define tensors, they ought to be "a way to multiply vectors that follows those rules". Quotienting is a generic technique to say "start with this object, and add this additional rule while keeping all of the others". Another way to say this is that the tensor algebra is the "free associative algebra": it's a way to multiply vectors where the only rules you have to reduce expressions are the ones you needed to have. [0] https://www.youtube.com/live/mqt1f8owKrU?t=500 https://www.youtube.com/live/mqt1f8owKrU?t=500 [1] https://en.wikipedia.org/wiki/Tensor_product#As_a_quotient_space https://en.wikipedia.org/wiki/Tensor_product#As_a_quotient_s...