I think so, too! Sometimes we should re-evaluate our expectations and think about what we really want and why we want it. Furthermore, I think it’s important to realise that others might have other motivations such as money, success, curiosity, altruism and so forth so there really is little need to compare ourselves to them.
Most of us don’t have the energy to be productive all the time. It took a lot of work (ironically) for me to be okay with that and to allow myself to do nothing at times. Procrastination for me was the result of constant pressure which sucked so much energy that I just couldn’t BE productive anymore. Funnily enough when I allowed myself to slack around or fall a bit behind I ended up greatly improving my grades.
In the end for me it’s a tale of starting to live in the moment (as cliche as that might sound) instead of failing to meet one’s own impossible expectations.
Also when it comes to applying what one learned by solving exercises or doing proofs feedback is so crucial and helpful. Firstly, one sees the parts one didn’t fully understand yet and secondly your way might be correct even if it isn’t the same way the provided solution approached the problem.
At the university I worked at we were still using punched cards (both punched and non-punched) to take written notes. When I left last year there was still a room full of those. Crazy that to think that at some point people took the decision to order huge amounts of these cards - maybe in the hopes that this technology will last for quite a long time.
What I don’t quite understand is why array was used here in the first place instead of equation* or similar.
Anyway interesting post I wouldn’t have guessed that carries onto the next array.
Why does it only affect the first cell (or first part of the first row) though instead of the whole row?
It was a third hand file with a lot of packages and the content was quite different, so perhaps there was a good reason in the original file that was lost in the minimization or perhaps the problem is that sometimes people prefer to reuse a few environments for all the possible constructions.
About the first cell: IIUC You can determine the color by the row, the column or by the individual cell, so the color is recalculated in each cell. For some reason, it fails in the first cell. I'm still curious.
While I agree that exercises etc. are the way to deeply understand the subject I disagree with your take that math lectures are senseless. I admittedly always struggled when I skipped lectures and attending them helped me to see the “bigger connections” and actually understand proofs given I was taking notes throughout the lecture.
In philosophy however - which for me this semester is entirely online without any video chat whatever interaction - the in person lectures were highly valuable to me due to their nature on heavily relying on discussion. While I do have a paper here and there to write my understanding mostly comes from attending the lectures.
The reason the matrix A has to be square for det(A-XI_n) is a) because we cannot subtract a matrix A from another matrix B if their dimensions aren’t the same and since XI_n is a nxn matrix A has to be an nxn matrix as well and b) because the determinant is only defined for square matrices.
https://en.m.wikipedia.org/wiki/Capital_ẞ