(405) What is a tensor product in simple words? – Quora

For any two vector spaces $V$ and $W$, the tensor product $V\bigotimes W$ is the space of bilinear functions on $V\times W$ .

If $V$ and $W$ are inner product spaces then for any $v\in V$ and $w\in W$ we can define the pure tensor $v\otimes w$ in $V\bigotimes W$ by $v\otimes w (v*,w*)=(v\cdot v* )(w\cdot w*)$ also often written by physicists as $\langle v|v*\rangle \langle w|w*\rangle$. But there are also elements of $V\bigotimes W$ that are not of the pure tensor form.

For example if $v_1\otimes w_1+v_2\otimes w_2$ could be written in the form $(a_1v_1+a_2v_2)\otimes(b_1w_1+b_2w_2)$, then for all $v\in V$ and $w\in W$, we’d need \begin{align}&(a_1v_1+a_2v_2)\otimes(b_1w_1+b_2w_2)(v,w)\\ &=(a_1v_1\cdot v)(b_1w_1\cdot w)+(a_1v_1\cdot v)(b_2w_2\cdot w)\\&+(a_2v_2\cdot v)(b_1w_1\cdot w)+(a_2v_2\cdot v)(b_2w_2\cdot w)\\&=(v_1\cdot v)(w_1\cdot w)+(v_2\cdot v)(w_2\cdot w)\end{align} .

But this is only true if $a_1b_1=a_2b_2=1$ and $a_1b_2=a_2b_1=0$, but if one of $a_1$ or $b_2$ is zero then one of $a_1b_1$ or $a_2b_2$ must be also.