Change of basis of tangent space Lee, Smooth Manifolds I have a question about the passage in chapter 11 of Lee's smooth manifold:


So if I am understanding this correctly, the matrix for the change of basis for the tangent space takes in vectors from our old coordinate and gives us vectors in terms of our new coordinate. Now traditionally (i,j) stands for (row, column) but I wrote the matrix out and this is impossible if that were the case. It only makes sense if (i,j) is actually (column, row). Is that the case here?
 A: You are right, if we write a matrix $A$ in the form $A = (a_{ij})$, then the first index $i$ denotes the row number in which we find the entry $a_{ij}$ and the second index $j$ denotes the column number. However, we can equally well write $A = (a_{ji})$ or $A = (a_{kl})$ or something else - simply because the two index variables do not have an intrinsic meaning.
To emphasize it again, the essential point is that the first index always denotes the row number and the second index always denotes the column number. This is of course merely a convention, but it is a commonly accepted convention and no author would be well-advised to ignore it. Also Lee complies with this convention. Perhaps it is a bit unusual that he uses the "$A = (a_{ji})$" notation, but it is completely legit.
Have a look at the computation of the matrix of $F_* : T_p\mathbb R^n \to  T_{F(p)}\mathbb R^m$ of a smooth map $F : U \to  V$ in terms of the
standard coordinate bases, where $U \subset \mathbb R^n$ and $V \subset \mathbb R^m$  are open. Lee obtains the formula
$$F_*\frac{\partial}{\partial x^i} \mid_p = \frac{\partial F^j}{\partial x^i}(p)\frac{\partial}{\partial y^j}\mid_{F(p)}$$
(recall the sum convention). This gives as the matrix of $F_*$
$$\mathcal M(F_*) =  \begin{pmatrix} \dfrac{\partial F^1}{\partial x^1}(p) \dots \dfrac{\partial F^1}{\partial x^n}(p)  \\ ... \\ \dfrac{\partial F^m}{\partial x^1}(p) \dots \dfrac{\partial F^m}{\partial x^n}(p) \end{pmatrix} \tag{1}$$
which is the usual Jacobian matrix of $F$. He writes it in the form
$$\mathcal M(F_*) = \begin{pmatrix} \dfrac{\partial F^j}{\partial x^i}(p) \end{pmatrix} \tag{2}$$
If we only look at $(2)$ we see that the entries of $\mathcal M(F_*)$ have two indices $i$ and $j$, but it is indeed not obvious which of $i$ and $j$ refer to rows and columns, respectively. We can even regard it as misleading if we assume the "$A = (a_{ij})$" notation.
However $(1)$ makes clear that Lee means
$$\mathcal M(F_*) = (a_{ji})$$
with $a_{ji} = \dfrac{\partial F^j}{\partial x^i}(p) $.
