What needs to be linear for the problem to be considered linear?

Question

Harry Altman presented an excellent question in a comment here: What needs to be linear for the problem to be considered linear? So is it enough to a have linear objective function or other requirements for example to the smoothness?

Answer 1

In a linear context, you have coefficients. In other words, you've got a module $M$ over a ring $R$. Then we can take linear combinations, $r_1m_1+\cdots +r_nm_n$, with $r_i\in R$, $m_i\in M$. Choosing a module, $M$, and a ring, $R$, let's you define linear in all kinds of contexts. Maybe you want $R=\mathbb{R}$ and $M$ to be a vector space. Then we can get systems of linear equations. Maybe $R$ and $M$ are $\mathcal{C}^\infty$ functions of variable $x$ for $R$ and variables $x$ and $y$ for $M$. This sets up a context for differential equations.