Identity/Matrix to basis/Transformation matrix/Example

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

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

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

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

directly

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

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