Identity/Matrix to basis/Transformation matrix/Example

Let ${\displaystyle {}V}$ denote a vector space with bases ${\displaystyle {}{\mathfrak {v}}}$ and ${\displaystyle {}{\mathfrak {w}}}$. If we consider the identity

${\displaystyle \operatorname {Id} \colon V\longrightarrow V}$

with respect to the basis ${\displaystyle {}{\mathfrak {v}}}$ on the source and the basis ${\displaystyle {}{\mathfrak {w}}}$ on the target, we get, because of

${\displaystyle {}\operatorname {Id} (v_{j})=v_{j}=\sum a_{ij}w_{i}\,,}$

directly

${\displaystyle {}M_{\mathfrak {w}}^{\mathfrak {v}}(\operatorname {Id} )=M_{\mathfrak {w}}^{\mathfrak {v}}\,.}$

This means that the describing matrix of the identical linear mapping with respect to the bases ${\displaystyle {}{\mathfrak {v}}}$ and ${\displaystyle {}{\mathfrak {w}}}$ is the transformation matrix for the base change from ${\displaystyle {}{\mathfrak {v}}}$ to ${\displaystyle {}{\mathfrak {w}}}$.