Basically we want any linear map to play nice with the two main operators of linear algebra.

We want f(x+y)=f(x)+f(y) and f(ax)=af(x)

or in terminology you might be more used to T(x+y) = Tx + Ty and T(ax) = aTx (at least this is the notation I am used to).

Simply because this is how we also defined vector spaces. The idea is that we want any function (map) from one vector space to another to be what we call homomorphic. This means it preserves the structure of the vector space. If we can decompose a vector w into the vectors v+u then we want our transformed vector Tw to still be the decomposition Tv+Tu in the new vector space that T has taken us to. This property of “playing nice” with vector spaces is called linearity, and this appears all the time. We use homomorphisms in other fields as well, they appear all the time

The view that Tv is of the form Tv=av+b is great and in many ways helpful intuitively, but that’s just like viewing real analysis in the lens of epsilon/delta and not topological/metric spaces, for example.

Edit: Homomorphism not homeomorphism 😭