Vector space/Tensor product/Bases/Transformation matrix/Fact/Proof

Proof

Due to the definition of the base change matrix, we have

${}v_{is}=\sum _{r\in J_{i}}a_{irs}w_{ir}\,.$

Therefore, using fact  (3), we obtain

${}{\begin{aligned}v_{1k_{1}}\otimes _{}\cdots \otimes _{}v_{nk_{n}}&={\left(\sum _{j_{1}\in J_{1}}a_{1j_{1}k_{1}}w_{1j_{1}}\right)}\otimes _{}\cdots \otimes _{}{\left(\sum _{j_{n}\in J_{n}}a_{nj_{n}k_{n}}w_{nj_{n}}\right)}\\&=\sum _{(j_{1},\ldots ,j_{n})\in J}a_{1j_{1}k_{1}}\cdots a_{nj_{n}k_{n}}w_{1j_{1}}\otimes _{}\cdots \otimes _{}w_{nj_{n}},\end{aligned}}$

and these coefficients constitute the base change matrix.

To fact