Homomorphism space/Bases/Matrices/Fact/Proof

The bijectivity was shown in fact. The additivity follows from

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

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