While posing another question I got stuck on the distinction of the following two concepts;
The components of a vector is usually referred to as it's coordinates, while trying to understand what "change of variables" means when it comes to systems of ODE's, I saw several sourse talked about change of variables as "a change of coordiinates".
Sure a function has a graph $(x,f(x))$
where if we do some kind of transformation we get a new graph. But this should be the same kind of coordinates as in a linear space.
Is this the thing that one refers to when one says "change of coordinates" as in change to polar coordiantes?
"Change of coordinates" comes up in many contexts in mathematics. Even in calculus, there are multiple contexts. For example, the graph $x^2+y^2=k^2$ is not a function in x. However, when we change from rectangular to polar coordinates, we get a function r = k. Then it's easier to compute tangent lines and area. In integral calculus, u-substitution is a "change of coordinates" to make integration easier.
In linear algebra, a change of coordinates comes up in linear transformations and diagonalization. In many contexts, we change the coordinates to make calculations easier. In exchange, there's a little bit of work to change the coordinates, either by using the Jacobian or finding a change of basis matrix.