TITLE: What is the difference between Cartesian and Tensor product of two vector spaces
QUESTION [23 upvotes]: In particular, how is it that dimension of Cartesian product is a sum of dimensions of underlying vector spaces, while Tensor product, often defined as a quotient of Cartesian product, has dimension which is a product of dimensions of underlying vector spaces.
Can this be easily visualized and explained in the case of RxR Cartesian product? 
How do the requirements of bilinearity and linearity cause a 'switch' from (x1,x2) to x1*x2?

REPLY [13 votes]: Given two vector spaces over the same ring $K$: $(V_{1},+_{1},*_{1})$, $(V_{2},+_{2},*_{2}).$ The Cartesian product of the two vector spaces is the new vector space $(V_{1}\times V_{2},+,*)$ with the new operations $(x,y)+(z,w)=(x+_{1}z,y+_{2}w),$ $\alpha*(x,y)=(\alpha*_{1}x,\alpha*_{2}y)$ for every $x,z\in V_{1},\;\;$ $y,w\in V_{2}$ and $\alpha\in K.$
The tensor product of the vector spaces $V_{1}$ and $V_{2}$ is the set of all linear (in both arguments) applications (mappings) from $V_{1}\times V_{2}$ onto $K;$ which means $V_{1}\otimes V_{2}=\{T:V_{1}\times V_{2}\mapsto K,\;\;T(\alpha x+\beta z,y)=\alpha*T(x,y)+\beta*T(z,y),\;\;T(x, \rho y+\sigma w)=\rho*T(x,y)+\sigma*T(x,w) \}.$
The difference between Cartesian and Tensor product of two vector spaces is that the elements of the cartesian product are vectors and in the tensor product are linear applications (mappings), this last are vectors as well but these ones applied onto elements of $V_{1}\times V_{2}$ gives a $K-$number.