I think this is a fantastic definition of what tensors are for mathematicians, but I think it is likely to be not very helpful for someone whose perspective is:

> Tensors... what the heck are those? Like matrices, but _more_ complicated, you say? Thanks, but I'll stick with my basic monoids over here. :-)

If matrices are the limit of your tolerance for complication, bilinear maps probably aren't intuitive tools for you.