As a reference, it looks very useful.
I thank the author for the slides, but this little proof need some more care, I don't know about the quality of other sections or the overall quality of the slides. Anyway I like how he tries to make things easy but good work is hard.
Edited: I was wondering whether a LLM reading Lecture 7 would detect what was missing in the proof. I tried with deepseek but its first feedback on the Lecture 7 was positive, then when prompted about the incomplete proof it recognized it as a common error and explained how to complete the proof. Also I have to prompt it about the bad factor 2 for it to detect it. So it seems that deepseek is not a useful tool to judge quality of math content without very expert guidance, deepseek suggested to ask the LLM to compare this proof with another proof to detect important or vital differences.
There aren't even any real details to fill in, you iterate on the lower right block so anything you do is orthogonal to the upper left block. Do a 2x2 block matrix multiplication to convince yourself that this preserves the form achieved so far.
I don't consider this a proof. Perhaps you have in mind two simple but key properties of reflections about the hyperplane orthogonal to a vector v: (a) The hyperplane of a reflection is the fixed point of the reflection (b) the hyperplane is the orthogonal vector space to the vector space spanned by v. From this two properties it follows that each step of making zeroes does not change previous zeroes.
Your claim that for advanced students there is no need to comment about details it is not falsifiable. Citing Mac Lane: A monad is just a monoid in the category of endofunctors.
But from a practical point of view one can see the very basic level and simplicity of the definitions and calculations prior to the proof. So at this level of detail I consider that noticing that one must be careful to not destroy previous zeros is matching the level of discourse at the proper level.
The proof says iterate on A, so that obviously creates a lower dimensional rotation Q that will act on the full space as above.
Absolutely mention this in lecture notes/during the lecture.
staplung•10mo ago
https://github.com/michiganrobotics/rob101/tree/main