In Analytical Mechanics, as physicist, we often use the expression $q_1, q_2$ are independent coordinates; however, this is a different concept than what we, as mathematicians, call linearly-independent vectors in a vector space.
The reason for that, first of all, since $q_i$'s are scalars, they just span a one-dimensional vector space over the real numbers, and secondly, what independent means in this concept is that none of the $q_i$'s is a function of other $q_i$'s, i.e for example, knowing the values of $q_1,q_2,...,q_N$ does not allow one to determine the value of $q_{N+1}$.
I was wondering is there any mathematical theory & field where this kind of relation between variables and relation between them are studies, similar to linear algebra, where we work with vectors, analyse the relation between them, and what we can do with them.
In Analytical Mechanics, you are dealing with the configuration space, which mathematically is a Riemannian manifold, and the $q_i$ are the coordinates in your manifold (they are called generalized coordinates in the Lagrangian context) and the kinetic energy defines the natural metric. On the tangent bundle to that manifold you have defined a function $L(q,q')$ (the Lagrangian) and you minimize the action,which is a functional defined on, for example, all paths on your manifold connecting two points. The mathematical theory behind all this is differential geometry and variational calculus on Riemannian manifolds. In the more symmetric Hamiltonian approach, you are dealing with the phase space (a manifold again) with a canonical symplectic structure on it. Thus, symplectic differential geometry is the mathematical framework in that case.