Why and How to Change Between Bases 

Thank you for the kind words! I am working on the next video, but it will take a while to finish

@@apastron11 One point that I think could be made a little more clearly is that is some vector with respect to the standard basis, which isn't mentioned in the video. The hardest thing about change of basis is not the computation, but keeping track of which basis we are using to express coordinates. I still struggle to internalize it; I find that your animation is a little quick at around 8:15, so I replay the segment several times to make sure I follow the manipulations you're making. They are, of course, correct. The matrix Q is then the augmented side of the RREF of matrix for beta augmented by the matrix for gamma: We start by assuming the bases are linearly independent. Cheers, again!

@@apastron11 Also, just after 10:05, my feeling is that you haven't adequately motivated the selection of the vectors in the basis set beta. In the preceding graphic, you showed that these directions are only scaled by operating on them with A. At least I think that's what you're showing. That would make these eigenvectors, and that is what we want when raising a matrix to a power. Please forgive me if I am only hopelessly confused by the way you have developed this step. I think this material would much better have been left for a subsequent presentation. It's a bit jarring the way you do it, because the concept of the eigenvectors seems to come out of nowhere. You didn't publish other videos to support this. It's as if you want to throw it in here because it's an important topic, but you should prepare the ground a bit more thoroughly.

@@danieljulian4676 ​​⁠​⁠ Yes, those vectors are eigenvectors, but I didn’t want to get too much into their selection because eigenvectors are a such big topic. I feel like if I were to get into the details of how those vectors were found it would distract from the main idea of the video. I just wanted to show a case of change of basis simplifying a problem. I do see how this could have made it more confusing, but I didn't know what else I could have done without getting too much into the details. I am considering a video about eigenvectors in the near future.

I have some other videos planned but I will look into these