User:IssaRice/Linear algebra/Geometry of linear transformations

There are various "geometric properties" that transformations can have, such as "preserves lengths", "preserves angles", and so forth. We might wonder, what kind of similar property could characterize linear transformations? Come to think of it, the definition often used for linear maps, that of T(v+w)=Tv + Tw, and T(av)=aTv, is very convenient from a symbolic-manipulation point of view, but gives hardly any idea what a linear map could mean.

For example, it's pretty obvious that linear maps preserve the origin. They also send lines to lines. They also send any equally-spaced-apart collinear points to equally-spaced-apart collinear points (not necessarily spaced apart at the same rate). They send lines through the origin to lines through the origin. They send parallel lines to parallel lines.

What's not obvious is what is the "minimal list" of properties that must be preserved, such that if a function preserves those properties, then it is necessarily linear. In other words, proving linear implies [list of properties] is easy, but proving the converse, [list of properties] implies linear, is harder. It's also hard to figure out which list of properties one ought to use.

(the above is all just motivation. when i feel like it i'll fill in the rest)