Linear mapping/Change of field/Matrix/Fact

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be an ${\displaystyle {}n}$-dimensional ${\displaystyle {}K}$-vector space, endowed with a basis ${\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}}$, and let ${\displaystyle {}W}$ be an ${\displaystyle {}m}$-dimensional ${\displaystyle {}K}$-vector space, endowed with of a basis ${\displaystyle {}{\mathfrak {w}}=w_{1},\ldots ,w_{m}}$. Let

${\displaystyle \varphi \colon V\longrightarrow W}$

be a linear mapping with the describing matrix ${\displaystyle {}M=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )={\left(a_{ij}\right)}}$. Let ${\displaystyle {}K\subseteq L}$ be a field extension.

Then the linear mapping

${\displaystyle \varphi _{L}\colon V_{L}\longrightarrow W_{L}}$

given by extension of scalars

is also described by the matrix ${\displaystyle {}M}$, considered over ${\displaystyle {}L}$, with respect to the bases ${\displaystyle {}1\otimes v_{1},\ldots ,1\otimes v_{n}}$ of ${\displaystyle {}V_{L}}$, and ${\displaystyle {}1\otimes w_{1},\ldots ,1\otimes w_{m}}$ of ${\displaystyle {}W_{L}}$.