Linear mapping/Matrix to bases/Correspondence/Fact/Proof

Proof

We show that both compositions are the identity. We start with a matrix ${}M={\left(a_{ij}\right)}_{ij}$ and consider the matrix

$M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)).$

Two matrices are equal, when the entries coincide for every index pair ${}(i,j)$. We have

${}{\begin{aligned}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)))_{ij}&=i-{\text{th coordinate of }}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M))(v_{j})\\&=i-{\text{th coordinate of }}\sum _{i=1}^{m}a_{ij}w_{i}\\&=a_{ij}.\end{aligned}}$

Now, let ${}\varphi$ be a linear mapping, we consider

$\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )).$

Two linear mappings coincide, due to fact, when they have the same values on the basis ${}v_{1},\ldots ,v_{n}$. We have

${}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )))(v_{j})=\sum _{i=1}^{m}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))_{ij}\,w_{i}\,.$

Due to the definition, the coefficient ${}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))_{ij}$ is the ${}i$-th coordinate of ${}\varphi (v_{j})$ with respect to the basis ${}w_{1},\ldots ,w_{m}$. Hence, this sum equals ${}\varphi (v_{j})$.

To fact