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 $𝔳$ and $𝔲$ be bases of $V$, and $𝔴$ and $𝔷$ bases of $W$. Let

$\phi :V⟶W$

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

Then $\phi$ is described with respect to the bases

$𝔲$ and $𝔷$ by the matrix

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

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

$𝔴$ to $𝔷$.

Proof, Write another proof