I'm trying to understand the concept of "coordinate independence" that is so prevalently used in physics, especially concerning tensors. The accepted answer here gives an example: Coordinate independence of geometrical objects.
"Some authors (often physicists) define everything in terms of coordinates and then show that the resulting objects are actually independent of the coordinates used to define them."
Most examples I've seen on wikipedia that demonstrate coordinate independence do it for things like linear operators. I don't understand this - as soon as we're talking about a function or mapping, is it not intrinsically something that is coordinate independent? $f(a+b) = f(c+d)$ if $a+b = c+d$ - it doesn't matter how we decompose the input, clearly the output will be the same.
Similarly, as soon as we're talking about an concrete vector in a concrete vector space, it seems to be that it's by definition something that is coordinate independent.
So I'm confused. Could I ask for a simple example of where this "coordinate" independence turned not to be satisfied?
I am not sure if this answers your question, but consider the isomorphism between a vector space $V$ and it's dual $V^*$ using dual bases.
Suppose $V$ was real, one dimensional, let $e$ be a non zero vector. Then $\{e\}$ and $\{2e\}$ are two different bases of $V$. Let $f_1, f_2$ be the corresponding duals. Now we have isomorphisms $T_1, T_2\colon V\to V^*$ such that $T_1(e) = f_1, T_2(e) = f_2/2$.
However, these two are not the same objects in $V^*$ because $f_1(e) = 1$ whereas $(f_2/2)(e) = 1/4$. The point is that the isomorphism between $V, V^*$ obtained using the dual basis is very much dependent on the choice of the basis on $V$.
One other situation where we need to make sure of coordinate independence is when defining functions on manifolds by patching up local definitions (defined using coordinates).