Vector space/Tensor product/Rules/Fact

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ denote vector spaces over ${}K$. Then the following calculation rules hold.

1. For vectors ${}v_{j}\in V_{j}$, and ${}c\in K$, we have

   $v_{1}\otimes _{}\cdots \otimes _{}v_{i-1}\otimes cv_{i}\otimes v_{i+1}\otimes _{}\cdots \otimes _{}v_{n}=c{\left(v_{1}\otimes _{}\cdots \otimes _{}v_{i-1}\otimes v_{i}\otimes v_{i+1}\otimes _{}\cdots \otimes _{}v_{n}\right)}\,.$

2. For vectors ${}v_{j}\in V_{j}$, we have

   ${}v_{1}\otimes _{}\cdots \otimes _{}v_{i-1}\otimes 0\otimes v_{i+1}\otimes _{}\cdots \otimes _{}v_{n}=0\,.$

3. Let ${}v_{i1},\ldots ,v_{im_{i}}\in V_{i}$, and ${}a_{ij}\in K$. Then

   ${}{\begin{aligned}{\left(\sum _{j=1}^{m_{1}}a_{1j}v_{1j}\right)}\otimes _{}\cdots \otimes _{}{\left(\sum _{j=1}^{m_{n}}a_{nj}v_{nj}\right)}&=\sum _{(j_{1},\ldots ,j_{n})\in \{1,\ldots ,m_{1}\}\times \cdots \times \{1,\ldots ,m_{n}\}}a_{1j_{1}}\cdots a_{nj_{n}}v_{1j_{1}}\otimes _{}\cdots \otimes _{}v_{nj_{n}}\\\,\end{aligned}}$

   holds.

Proof, Write another proof