I am self-studying Lee's Introduction To Smooth Manifolds, and I want to make sure I am understanding a point correctly. When discussing the differential $dF_{p}:T_{p} \rightarrow T_{F(p)}N$ of a smooth function between manifolds, there is a nice characterization of it's matrix representation, which is given by the following:
If $(U, \gamma)$ and $(V,\psi)$ are charts such that $p \in U$, $F(U)\subseteq V$, and
$\hat{F}=\psi \, \circ F \, \circ \gamma^{-1} : \gamma(U \cap F^{-1}(V)) \rightarrow \psi(V) $
is smooth, then the matrix representation of $d_{p}F$ is given by the Jacobian of $\hat{F}$ evaluated at $\psi(p)$, or $D\hat{F}(\psi(p))$.
Now, my question is the following: We know that a linear map is invertible if and only if it's matrix representation is. I would like to apply the same logic to the above to the linear map $d_{p}F$, but it seems that it's matrix representation was dependent on the charts we began with. Is it true that $d_{p}F$ is invertible if and only if $D\hat{F}(\psi(p))$ is, regardless of the charts we choose?
Yes, it is true that $d_pF$ is invertible if and only if $D\widehat{F}(\psi(p))$ is, by very much the same reason as to why a linear map is invertible if and only if a given matricial representation of it is. The expression of $d_pF$ as a matrix (the Jacobian) given charts $\gamma, \psi$ is just the expression of the linear map $d_pF$ with respect to the bases $(\partial_i(\cdot \circ \gamma^{-1}))_{i=1,\cdots ,n}$ and $(\partial_i(\cdot \circ \psi^{-1}))_{i=1,\cdots ,n}$ of $T_pM$ and $T_{F(p)}N$, respectively.
The fact that the matrix depends on the choice of charts is just a result of the fact that different charts furnish different associated bases of the tangent spaces. The situation is just plain linear algebra, the charts are just used in order to choose the bases.