I have some perplexities when I reading references about tensor product.
My main question is:
How to define the tensor product between two vectors?
It is clearly to define the tensor product of two column vectors as $v_1\otimes v_2=v_1v_2^\top$ which is according to matrix multiply (see Matrix multiply).
What is more, to define the tensor product of two linear maps, we have: $$ (A_1\otimes A_2)(v_1\otimes v_2)=w_1\otimes w_2 \qquad(1) $$ where $A_1 v_1=w_1$ and $A_2 v_2=w_2$ (see tensor product). When $A_1$ and $A_2$ are maps between column vectors, $(A_1\otimes A_2)$ can be written as Kronecker product.
However, these definitions confuse me. For example, let $v_1$,$v_2$, $w_1$ and $w_2$ are $2\times1$ column vectors and then $A_1$ and $A_2$ are $2\times$ matrices. They satisfy $A_1 v_1=w_1$ and $A_2 v_2=w_2$. Obviously, the Kronecker product $A_1\otimes A_2$ is a $4\times4$ matrix while $v_1\otimes v_2$ and $w_1\otimes w_2$ are all $2\times2$ matrices. Then $(A_1\otimes A_2)(v_1\otimes v_2)$ is meaningless because we can not calculate the multiply of $4\times4$ matrix and $2\times 2$ matirx ! So how to check $(1)$?
In 'Alexander Graham, Kronecker Products and Matrix Calculus: with Applications', the author check $(1)$ by define $v_1\otimes v_2$ and $w_1\otimes w_2$ as Kronecker product which is different from $v_1v_2^\top$.
So, as mentioned in beginning, how to define the tensor product between two vectors? Is there do not exist a unique definition about it? Or there is some points I missed?
Thanks for help!