What is linearity?
The definition of linearity depends on context.
- A linear map satisfies the conditions above.
- A linear DE means that the associated Differential operator is linear in each derivative of the unknown.
- The solutions to a linear equation are the roots of an affine linear map.
- Linear algebra deals with vector spaces and (affine) linear maps.
- Linear programming is about linear objective functions and affine constraints.
- Linear interpolation is interpolation of a function by an affine linear map.
The term affine linear used here is defined by: $f:X\to Y$ is affine linear iff there exists $a\in Y$ such that $x\mapsto f(x)-a$ is linear, i.e. $f(x) = g(x) + a$ where $g$ is linear.
My answer is that linearity, in your examiner's perspective, is a canonical function between structures $X\rightarrow Y$with a commutative '$+$' and an distributive action '$\cdot$': $a\cdot(x+y)=a\cdot x + a\cdot y$. The function is such that the diagram commutes: $\require{AMScd}$ \begin{CD} A\times X\times X @>(1,f,f)>> A\times Y\times Y\\ @V S_X V V\# @VV S_Y V\\ X @>>f> Y \end{CD} That is, the function should satisfy $S_Y(1,f,f)=fS_X$. This gives the condition $S_Y((1,f,f)(a,x,y))=f(S_X(a,x,y))\Leftrightarrow S_Y(a,f(x),f(y))=f(a\cdot(x+y))\Leftrightarrow$ $a\cdot(f(x)+f(y))=a\cdot f(x) + a\cdot f(y)=f(a\cdot(x+y))$.
This seems to be possible to extend to all mathematical structures.
Linearity in your perspective perhaps referring to the lack of nonlinear variable terms.
On a "philosophical" level : a process is linear if when you double the input, you will also double the output.
The rigorous mathematical definition of linear maps, linear differential equation, etc. have already been given.
The whole idea of differential calculus is that when you zoom enough, anything looks linear at small scale : that's what a derivative is.