What is the difference between Cartesian and Tensor product of two vector spaces

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?

Solutions Collecting From Web of "What is the difference between Cartesian and Tensor product of two vector spaces"

The tensor product of two vector spaces is not a quotient of the Cartesian product of those spaces. It is a quotient of the free vector space with basis the cartesian product. That is, $V \otimes W$ is a quotient of an enormous infinite dimensional vector space. A vector space with basis
$$\{x_\alpha \ | \ \alpha \in V \times W\},$$
so there is one basis element for each element of $V \times W$ (hence enormous).

If you fix bases $\{v_i\}$ and $\{w_j\}$ of $V$ and $W$ then, because of the relations that one quotients by, the tensor product has as basis those $\overline{x_\alpha}$ (where the overline means the element of the quotient represented by $x_\alpha$) with $\alpha = (v_i, w_j)$ for some $i, j$. There are $\dim V$ choices for $v_i$ and $\dim W$ choices for $w_j$ hence there are $(\dim V)(\dim W)$ choices for $\alpha$. Thus $\dim(V \otimes W) = (\dim V)(\dim W)$.

On the other hand the cartesian product $V \times W$ has basis
$$\{(v_i, 0), (0, w_j) \ | \ \text{for all values of} \ i, j\}$$
and there are $\dim V + \dim W$ elements in that set so $\dim(V \times W) = \dim V + \dim W$.