Change of basis vs change of coordinate system

Question

To my understanding, a change of coordinate system can be thought of as any set of invertible relations between the old coordinates and the new. i.e. every new coordinate can be represented as a function of the old coordinates and vice versa, regardless of whether that function is linear or not. This means that a change of coordinates cannot be generally thought of as a change of basis (invertible linear transformation). Is that correct?

I am asking this because I had been learning about tensors, and I noticed how sometimes they are defined with regard to how their components vary under change of basis, and other times they are defined with regard to how they vary under change of coordinates.

Thanks!

Answer 1

You compose two coordinate system $\phi_1:\Omega_1\to M$ and $\phi_2:\Omega_2\to M$ to define a change of coordinates in $M$, where the $\Omega_i$ are open sets of some $\Bbb{R}^n$, if $\psi=\phi^{-1}_2\circ\phi_1:\Omega_1\to\Omega_2$ then to describe vector space matters, on the tangent of $M$ at point $p$, we take $\phi_1=\phi_2\circ\psi$ and its derivative $J\phi_1=J\phi_2\cdot J\psi$ or, more precisely, its evaluated version $$J\phi_1|_a=J\phi_2|_{\psi(a)}\cdot J\psi|_a,$$ where $\phi_1(a)=\phi_2(b)=p$ and $\psi(a)=b$. The very same matrix $J\psi|_a$ maps (isomorphically) $$J\psi|_a:T_{p,\phi_1}M\to T_{p,\phi_2}M,$$ between the two tangent description of the tangent space $T_pM$ of $M$ at $p$.

When $M$ is a vector space the $J\psi$ is an invertible linear transformation or also a basis change.

Answer 2

This means that a change of coordinates cannot be generally thought of as a change of basis (invertible linear transformation). Is that correct?

Yes, a change of coordinates is not always simply a change of basis: take $\mathbb R ^2$ as $\mathbb R$-vector space and consider a simple translation by $b\in \mathbb R ^2\setminus \{0\}$ as the change of coordinate system, say from $B$ to $B'$, we denote this map with $\phi:\mathbb R ^2 \rightarrow \mathbb R ^2, x\mapsto x+b$

This map is not linear as $\phi (0) = b\neq 0$ (for $0 := 0_{\mathbb R ^2}:=(0,0)$), hence it cannot be represented as change of basis matrix (or this kind of tensor representation, since it does not preserve the structure of the space), but still being bijective and hence (only in the sense of function between sets, invertible). Seeing this as special case of tensor representation should answer your question.