Tensors: Acting on Vectors vs Multilinear Maps I have the feeling like there are two very different definitions for what a tensor product is.  I was reading Spivak and some other calculus-like texts, where the tensor product is defined as
$(S \otimes T)(v_1,...v_n,v_{n+1},...,v_{n+m})= S(v_1,...v_n) * T(v_{n+1},...,v_{n+m}) $
The other definition I read in a book on quantum computation, its defined for vectors and matrices and has several names, "tensor product, Kronecker Product, and Outer product": http://en.wikipedia.org/wiki/Outer_product#Definition_.28matrix_multiplication.29
I find this really annoying and confusing.  In the first definition, we are taking tensor products of multilinear operators (the operators act on vectors) and in the second definition the operation IS ON vectors and matrices.  I realize that matrices are operators but matrices aren't multilinear.  Is there a connection between these definitions?  

I can't comment yet (or I don't know how to if I can), but echo Thomas' response and want to add one thing.  
The tensor product of two vector spaces (or more generally, modules over a ring) is an abstract construction that allows you to "multiply" two vectors in that space.  A very readable and motivated introduction is given in Dummit & Foote's book.  (I always thought the actual construction seemed very strange before reading on D&F -- they manage to make it intuitive, motivated as trying to extend the set of scalars you can multiply with).
The collection of $k$-multilinear functions on a vector space is itself a vector space -- each multilinear map is a vector of that space.  The connection between the two seemingly different definitions is that you're performing the "abstract" tensor product on those spaces of multilinear maps.
It always seemed to me that the tensor product definition in Spivak was a particularly nice, concrete example of the more general, abstract definition.
