# Signature for vector spaces

Find a signature for vector spaces such that the homomorphisms correspond to linear maps.

My attempt: Let $$F$$ denote the corresponding field and consider the structure $$\{0;+,-,c_F\}$$, where $$0$$ is a constant, $$+$$ is a binary function symbol and $$-$$ and $$c_F$$ are unary function symbols (for each $$c\in F$$).

But I don't know how to show the following:

Homomorphisms correspond to linear maps.

I think I understand it intuitively but since I'm new to model theory, I don't know how to precisely write it.

A homomorphism between structures $$X,Y$$ of the given signature would be a function $$f:X\to Y$$ which satisfies $$f(0)=0,\quad f(x_1+x_2)=f(x_1)+f(x_2),\quad f(-x)=-f(x),\quad f(c_Fx)=c_F\,f(x)$$ for all $$x,x_1,x_2\in X$$ and $$c\in F$$.
If $$X,Y$$ happens to be vector spaces (with the usual interpretation of the operation symbols) then this amounts exactly to $$f$$ being a linear map.
By the way, in that case it suffices that $$f$$ preserves addition and scalar multiplication, then the vector space axioms ensure that it preserves zero and subtraction as well.