Cartesian & Tensor Product

Question

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?

Answer 1

The Cartesian product $V_1\times V_2$ consists of the set of all vectors $(v_1,v_2)$ with $v_1\in V_1, v_2\in V_2$ with component-wise addition and multiplication.

The tensor product $V_1\otimes V_2$ on the other hand is much larger. It consists of sums of elements of the form $v_1\otimes v_2$ with $v_1\in V_1,v_2\in V_2$. The addition is such that $v_1\otimes v_2+v_1'\otimes v_2=(v_1+v_1')\otimes v_2$, and similarly for the other coordinate, but in general $v_1\otimes v_2+v_1'\otimes v_2'$ cannot be written in the form $v_1''\otimes v_2''$.

Answer 2

Just adding on to the answer by @Alex Becker

You can moreover check that if $V$ and $W$ are finite dimensional, $\dim V\times W = \dim V + \dim W$, $\dim V\otimes W = (\dim V )(\dim W)$.

To really know how the two are different, I suggest an article by Tim Gowers (https://www.dpmms.cam.ac.uk/~wtg10/tensors3.html) named "How to Lose Your Fear of Tensor Products." It may take a little time to read through it, but I can promise you that you really will lose any fear of tensor products afterwards.