I have encountered an interesting equation while reading Spivak's Comprehensive Introduction to Differential Geometry (vol. 1, chapter 4).
The situation is as follows: we are considering coordinate systems on $\mathbb{R}^n$ which are linear transformations. Let $x$ be such a coordinate system with $x(v_i)=e_i$ (of course $x(a^1v_1+\ldots+a^n v_n)=(a^1,\ldots,a^n)$ ). Now, if $x'$ is another such coordinate system, then by means of linear algebra we clearly have $ {x'}^{j} = \sum_{i=1}^{n} a_{ij}x^{i} $ for certain $a_{ij}$.
The non-trivial part starts when the author gives the explicit formula for coefficients: $a_{ij}\;=\; \frac{\partial{x'}^{j}}{\partial{x}^{i}}$, so that ${x'}^{j}=\sum_{i=1}^{n} \frac{\partial {x'}^{j}}{\partial x^{i}} \: x^{i}$. It is even more disturbing since the author remarks on this result as somehow obvious.
I would be grateful for any hints on how to obtain such a formula for $a_{ij}$. Also, how should I understand it in terms of tangent spaces?
Thanks in advance
Take the equality $x'^j = \sum_{i=1}^n a_{ij}x^i$, and apply $\frac{\partial}{\partial x^k}$ on both sides. We get $$\frac{\partial x'^j}{\partial x^k} = \frac{\partial}{\partial x^k}\sum_{i=1}^n a_{ij}x^i = \sum_{i=1}^n a_{ij}\frac{\partial x^i}{\partial x^k} = \sum_{i=1}^n a_{ij}\delta^i_k = a_{kj}.$$Rename $k \to i$ if you want to get $\frac{\partial x'^j}{\partial x^i} = a_{ij}$.