As far as I can tell, it's only a slight exaggeration to say that every text has a different notation for a change of basis matrix from (say) $\mathcal{B}$ to $\mathcal{C}$. That's not even to talk about the "standard" matrix for representation e.g. changing from $\mathcal{B}$ to the standard basis $\mathcal{E}$ in $\mathbb{R}^n$.

My question is whether anyone has any ideas for why a particular notation is superior in the classroom (or other educational outlet). I'm not asking for lists of actual notation - that would be more appropriate for MSE anyway. I want specific notation that you feel in your experience helps students understand this notably thorny concept. (Yes, if you wrote a book, feel free to explain why you chose that notation!)

(Because seriously, at this point I need a dictionary just to translate from one text to another. A web search just now showed three or four notations I'd never even seen before, and I teach this course regularly!)

As far as I can tell, it's only a slight exaggeration to say that every text has a different notation for a change of basis matrix from (say) $\mathcal{B}$ to $\mathcal{C}$. That's not even to talk about the "standard" matrix for representation, e.g., changing from $\mathcal{B}$ to the standard basis $\mathcal{E}$ in $\mathbb{R}^n$.

My question is whether anyone has any ideas for why any particular notation is superior in the classroom (or other educational outlet) to other such notations. I'm not asking for a list of actual notations currently used - that would be more appropriate for math.se anyway.

I want to know the specific notation that you feel, in your experience, helps students understand this notably thorny concept. (Yes, if you wrote a book, feel free to explain why you chose that notation!)

Aside: Seriously, at this point, I need a dictionary just to translate from one text to another. A web search just now showed three or four notations I'd never even seen before, and I teach this course regularly!