My textbook says the following in an appendix on tensor notation:
Consider a change of coordinate axes in which the basis vectors $\mathbf{e}_i$ are replaced by a new basis set $\hat{\mathbf{e}}_j$, where $\hat{\mathbf{e}}_j = \sum_{i} H_j^i \mathbf{e}_i$, and $H$ is the basis transformation matrix with entries $H_j^i$. If $\hat{\mathbf{\mathrm{x}}} = (\hat{x}^1, \hat{x}^2, \hat{x}^3)^T$ are the coordinates of the vector with respect to the new basis, then we may verify that $\hat{\mathbf{\mathrm{x}}} = H^{-1}\mathbf{\mathrm{x}}$. Thus, if the basis vectors transform according to $H$ the coordinates of points transform according to the inverse transformation $H^{-1}$.
I don't understand what the last part is saying (and yes, I've made sure to copy it exactly):
Thus, if the basis vectors transform according to $H$ the coordinates of points transform according to the inverse transformation $H^{-1}$.
I have 2 problems with this part:
- It seems to me that the syntax is poor, which makes it difficult to try and understand what it is trying to say. Even so, I think I have managed to decipher it (see 2).
- Beyond the syntax and with regards to mathematics, I'm wondering why is it that we use $H$ as the transformation matrix for the basis vectors in $\hat{\mathbf{e}}_j = \sum_{i} H_j^i \mathbf{e}_i$, whereas we use $H^{-1}$ as the transformation matrix for the coordinates of points $\hat{\mathbf{\mathrm{x}}} = H^{-1}\mathbf{\mathrm{x}}$? In other words, I don't understand why we use $H$ for one and $H^{-1}$ for the other? I only have introductory-level linear algebra knowledge, so please explain gently.
I would greatly appreciate it if people could please take the time to help me understand this.