I understand that the standard basis, for $\mathbb{R}^3$, $\{\textbf{e}_1, \textbf{e}_2\, \textbf{e}_3\}$ are columns of $I_3$, and are illustrated in our book like this:
However, I am not sure how this relates to what we would call "x", "y", and "z". I understand that these labels for each of the basis vectors are not necessary for the problems and proofs we are doing in class, but I'm finding this an issue in a personal project.
Consider the following image
Focusing just on the 3 vectors $X_c$, $Y_c$, and $Z_c$, and ignoring the rest of the diagram, it's clear that the points plotted with respect to this basis are non standard. For example, the vector labeled $Y_c$ increases as it "goes down".
A problem I am currently experiencing is that I have some data in $\mathbb{R}^3$ that is measured with respect to the basis $\{X_c, Y_c, Z_c\}$ and I want to plot those points with respect to the standard basis $\mathbb{R}^3$, $\{\textbf{e}_1, \textbf{e}_2\, \textbf{e}_3\}$. When we do change of basis in linear algebra class, there is some kind of imposed order on each of bases, which makes it easy to find the coordinates of one basis vector in the basis of its corresponding basis vector.
For the problem presented above, how do I know the order of each of the bases? For the coordinate system determined by the basis $\{X_c, Y_c, Z_c\}$ there is a clear ordering based on the alphabet. However, how do I know which standard basis vectors $\{\textbf{e}_1, \textbf{e}_2\, \textbf{e}_3\}$ correspond to $\{X_c, Y_c, Z_c\}$?
Thanks for reading!