> Don't forget duality. A matrix can represent a linear map (or, if you prefer, a module homomorphism) but if you think of it as a map from V to W* instead of to W, it represents instead a bilinear form on VxW.
Okay, but why does that need a matrix to conceptualize? Given any linear map L: V → W, and another map φ: W → W* (which is equivalent to assuming a basis for W*, i.e. what you're doing when treating L as a map from V to W* instead of to W), then you can construct a bilinear form 〈v,w〉 = L(v)φ(w).
You don't need matrices to conceptualize these things, you need them to represent them for purposes of calculation. If you want to do calculations on linear maps V -> W, you usually do it with matrices. And if you want to do calculations on bilinear forms on VxW, you also usually do it with matrices.
Okay, but why does that need a matrix to conceptualize? Given any linear map L: V → W, and another map φ: W → W* (which is equivalent to assuming a basis for W*, i.e. what you're doing when treating L as a map from V to W* instead of to W), then you can construct a bilinear form 〈v,w〉 = L(v)φ(w).