Why is the tensor product constructed in this way? I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that we want is to have tuples of vectors that behave linearly on addition and multiplication by scalar, couldn't we just take all vector spaces $L_1, L_2,\dots,L_p$, form their cartesian product $L_1\times L_2\times \cdots \times L_p$ and simply introduce operations analogous to that of $\mathbb{R}^n$ ? 
We would get a space of tuples of vectors on wich all those linear properties are obeyed. What's the reason/motivation to define the tensor product using the free vector space and that quotient to impose linearity ? Can someone point me out the motivation for that definition ?
Thanks very much in advance.

When I studied tensor product, I am lucky to find this wonderful article by Tom Coates. Starting with the very trivial functions on the product space, he explains the intuition behind tensor products very clearly.
