Linear mapping/Change of field/Matrix/Fact/Proof

Because of fact, the given families are indeed bases. The basis element ${\displaystyle {}v_{j}}$ of ${\displaystyle {}V}$ is mapped under ${\displaystyle {}\varphi }$ to

${\displaystyle {}\varphi (v_{j})=\sum _{i=1}^{n}a_{ij}w_{i}\,.}$

Therefore, the basis element ${\displaystyle {}1\otimes v_{j}}$ of ${\displaystyle {}V_{L}}$ is mapped under ${\displaystyle {}\varphi _{L}}$ to

${\displaystyle {}\varphi (1\otimes v_{j})=1\otimes \varphi (v_{j})=1\otimes {\left(\sum _{i=1}^{n}a_{ij}w_{i}\right)}=\sum _{i=1}^{n}a_{ij}{\left(1\otimes w_{i}\right)}\,.}$

The coefficients ${\displaystyle {}a_{ij}}$ constitute also the describing matrix of ${\displaystyle {}\varphi _{L}}$.