Linear mapping/Finite dimensional/Change of basis/Fact

Let ${}K$ denote a field, and let ${}V$ and ${}W$ denote finite-dimensional ${}K$-vector spaces. Let ${}{\mathfrak {v}}$ and ${}{\mathfrak {u}}$ be bases of ${}V$ and ${}{\mathfrak {w}}$ and ${}{\mathfrak {z}}$ bases of ${}W$. Let

$\varphi \colon V\longrightarrow W$

denote a linear mapping, which is described by the matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )$ with respect to the bases ${}{\mathfrak {v}}$ and ${}{\mathfrak {w}}$.

Then ${}\varphi$ is described with respect to the bases

${}{\mathfrak {u}}$ and ${}{\mathfrak {z}}$ by the matrix

$M_{\mathfrak {z}}^{\mathfrak {w}}\circ (M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))\circ (M_{\mathfrak {u}}^{\mathfrak {v}})^{-1},$

where ${}M_{\mathfrak {u}}^{\mathfrak {v}}$ and ${}M_{\mathfrak {z}}^{\mathfrak {w}}$ are the transformation matrices, which describe the change of basis from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$ and from

${}{\mathfrak {w}}$ to ${}{\mathfrak {z}}$.

Proof, Write another proof