Definition of linear/ affine/ etc transformation

Question

A linear transformation is a function $f:V\to W$ from vector space $V$ to vector space $W$ such that linear combinations (alternatively vector addition and scalar multiplication) are preserved.

An affine transformation is similarly a between from an affine space to another such that all affine combinations are preserved.

Could we use this as an alternate definition: A linear(/ affine/ etc) transformation is a function from one l(/a/e) space to another such that the image of the function is also a l(/a/e) space?

Answer 1

Let define the following map: $$f:\left\{\begin{array}{ccc}\mathbb{R} & \rightarrow & \mathbb{R}\\x & \mapsto & x^3\end{array}\right..$$ $\mathbb{R}$ is a vector space, $f(\mathbb{R})=\mathbb{R}$ is a vector space, howeover, $f$ is not a linear map. Hence, unless I misunderstood your definition, you can not define linear maps this way.

Answer 2

Not exactly in the way that you formulate it, because it is not clear what the general meaning of the term "image of the function" would be.

But there is a general notion of "structure-preserving mapping". The abstract branch of mathematics that deals with these very general concepts is called category theory. The usual terms are "objects" (for the spaces) and "morphisms" (for the mappings).

Other examples of structure-preserving maps (beside the ones you quoted) include: continuous functions between topological spaces; isometries between metric spaces; different kinds of homomorphisms between various types of algebraic structures (groups, rings, rings with unit) etc.