This is kind of a follow-up to the excellent answer to this question: https://physics.stackexchange.com/questions/445948/general-coordinate-transformations
I want a very clear understanding of what all is going on under the hood. Let's just assume the manifold we're working with right now is just $\mathbb{R}^2$. When we do a linear transformation of a rectilinear coordinate system, it's generally taught that vector components transform like this:
$$A'^x=\frac{\partial x'}{\partial x}A^x + \frac{\partial x'}{\partial y}A^y\ \ ;\ \ \ \ \ \ \ \ A'^y=\frac{\partial y'}{\partial x}A^x + \frac{\partial y'}{\partial y}A^y$$
For a transformation from the Cartesian system to a general curvilinear system, from what I understand, instead of talking about how the components of $\mathbf{A}$ are related via coordinate system transformation, we have to talk about how the components in the tangent/cotangent spaces of a particular point $P$ in the $\mathbb{R}^2$ manifold are affected by the coordinate transformation.
Is the above understanding correct?
Assume that the coordinate system transformation is defined by $x^i=x^i(\mathbf{x}')$ and its inverse $x'^i=x'^i(\mathbf{x})$. Suppose we have some vector with representation $\mathbf{A}=\sum_iA^i\mathbf{e}_i$ before and $\mathbf{A}=\sum_iA'^i\mathbf{e}'_i$ after, where $\{\mathbf{e}'_i\}_i$ is the basis of the tangent space at some point $P$ in the transformed coordinate system defined as:
$$\mathbf{e}'_i=\bigg(\frac{\partial\mathbf{r}}{\partial x'^i}\bigg)_P$$
where the derivatives are evaluated at $P$. Similarly the basis $\{\mathbf{e}_i\}$ of the tangent space at $P$ in the initial coordinate system is:
$$\mathbf{e}_i=\bigg(\frac{\partial\mathbf{r}}{\partial x^i}\bigg)_P$$
To find $A'^j$, for example, I'll need to pick the covector $\mathbf{e}'^j$ and act it on $\mathbf{A}$:
$$A'^j=\mathbf{e}'^j(\mathbf{A})=\mathbf{e}'^j\bigg(\sum_iA^i\mathbf{e}_i\bigg) =\sum_iA^i\mathbf{e}'^j(\mathbf{e}_i)=\sum_iA^i\mathbf{e}'^j\bigg(\frac{\partial\mathbf{r}}{\partial x^i}\bigg) \\=\sum_iA^i\mathbf{e}'^j\bigg(\frac{\partial\big(\sum_kx'^k\mathbf{e}'_k\big)}{\partial x^i}\bigg) =\sum_iA^i\bigg(\frac{\partial\big(\sum_kx'^k\mathbf{e}'^j(\mathbf{e}'_k)\big)}{\partial x^i}\bigg) \\=\sum_iA^i\bigg(\frac{\partial\big(\sum_kx'^k\delta^j_k\big)}{\partial x^i}\bigg) =\sum_iA^i\frac{\partial x'^j}{\partial x^i}$$
I'd just like to know if the statements in bold in the above paragraphs are correct, or if there's a technical mistake or something I missed.
Note that I've deliberately avoided dot product usage in favor of covectors acting on vectors, just to explicitly understand the interplay of tangent/cotangent spaces.