Below I have added the relevant parts of the lecture notes that are necessary for the proof:
I have to check that the map $I$ is bijective but I don't know how they did it here, can somebody explain it? I only not understand the bijectivity part of the proof but I could understand why the map is linear so I did not add it in the picture.
EDIT: What I don't understand about this proof is that it says for every linear map there is a family and given a linear map there is only one corresponding unique family. Apparently this person talks about a map that goes from $\{a_{(j,i)}\in \text{F}:j\in\{1,...,m\}\times\{1,...,n\}=\mathcal{M}(m,n,F)$ to $\mathcal{L}(V,W)$ with $(a_{j,i})\mapsto \sum_{j=1}^{m}\sum_{i=1}^{n}a_{ji}A_{v_{i}w_{j}}$ but this map cannot be $I$ because the domain and the codomain does not match up. Please help me, I am really confused
coordinates
