Linear mapping/Finite dimensional/Change of basis/Fact/Proof

Proof

The linear standard mappings $K^n\to V$ and $K^m\to W$ for the various bases are denoted by ${\Psi }_{𝔳},{\Psi }_{𝔲},{\Psi }_{𝔴},{\Psi }_{𝔷}$. We consider the commutative diagram

$\begin{array}{ccccccc}{K}^n& & & \stackrel{M_{𝔴}^{𝔳}(\phi )}{⟶}& & & K^m\\ & \searrow {\Psi }_{𝔳}& & & & {\Psi }_{𝔴}\swarrow & \\ M_{𝔲}^{𝔳}\downarrow & & V& \stackrel{\phi }{⟶}& W& & \downarrow M_{𝔷}^{𝔴}\\ & \nearrow {\Psi }_{𝔲}& & & & {\Psi }_{𝔷}\nwarrow & \\ K^n& & & \stackrel{M_{𝔷}^{𝔲}(\phi )}{⟶}& & & K^m,\end{array}$

where the commutativity rests on fact and fact. In this situation, we have altogether

$\begin{array}{cc}{M}_{𝔷}^{𝔲}(\phi )& ={\Psi }_{𝔷}^{-1}\circ \phi \circ {\Psi }_{𝔲}\\ & ={\Psi }_{𝔷}^{-1}\circ ({\Psi }_{𝔴}\circ M_{𝔴}^{𝔳}(\phi )\circ {\Psi }_{𝔳}^{-1})\circ {\Psi }_{𝔲}\\ & =({\Psi }_{𝔷}^{-1}\circ {\Psi }_{𝔴})\circ M_{𝔴}^{𝔳}(\phi )\circ ({\Psi }_{𝔳}^{-1}\circ {\Psi }_{𝔲})\\ & =({\Psi }_{𝔷}^{-1}\circ {\Psi }_{𝔴})\circ M_{𝔴}^{𝔳}(\phi )\circ ({\Psi }_{𝔲}^{-1}\circ {\Psi }_{𝔳}{)}^{-1}\\ & =M_{𝔷}^{𝔴}\circ M_{𝔴}^{𝔳}(\phi )\circ (M_{𝔲}^{𝔳}{)}^{-1}.\end{array}$

To fact