Homomorphism space/Bases/Matrices/Fact/Proof

Proof

The bijectivity was shown in fact. The additivity follows from

${}{\left(M_{\mathfrak {w}}^{\mathfrak {v}}(f+g)\right)}_{ij}=((f+g)(v_{j}))_{i}=(f(v_{j})+g(v_{j}))_{i}=f(v_{j})_{i}+g(v_{j})_{i}={\left(M_{\mathfrak {w}}^{\mathfrak {v}}(f)\right)}_{ij}+{\left(M_{\mathfrak {w}}^{\mathfrak {v}}(g)\right)}_{ij}\,,$

where the index ${}i$ denotes the ${}i$-th component with respect to the basis ${}{\mathfrak {w}}$.

To fact