I just selected lecture 07 to take a look: Lecture 07 is about QR factorizacion and Householder reflections. The author proves how to construct a reflection to make zeros in the first column and then he just claims that following this procedure for the other columns finish the proof. But he should prove or justify why the other reflections do not destroy the zeros of previous reflections. Also he proves that a vector v is the vector to construct the reflection (but there is a factor of 2 that was not correctly simplified, maybe a latex error), but I think that it should be more general and easier to prove that for any w the vector from w to its image f(w) is the orthogonal vector to the plane of the reflection.
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.
That's an absolutely obvious step though? As in, detailed lecture notes should maybe elaborate with a sentence, but in a lecture I would not put this on the slides but mention the core point and expect students at this level (who should have seen some amount of more theoretical LinAlg courses by then) to understand how to do the 1 line calculation.
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.
-- 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.
I guess the title would better be "Numerical Linear Algebra Class in Julia at TUM". I.e. the "TUM" in the title does not mean that there's some new "TUM" version of Julia, rather that the class is at the Technical University of Munich.
This is a nicely comprehensive course, but it looks like it is pretty fast paced, especially in the last few lectures (some of those later slides definitely aren't finished).
I just selected lecture 07 to take a look: Lecture 07 is about QR factorizacion and Householder reflections. The author proves how to construct a reflection to make zeros in the first column and then he just claims that following this procedure for the other columns finish the proof. But he should prove or justify why the other reflections do not destroy the zeros of previous reflections. Also he proves that a vector v is the vector to construct the reflection (but there is a factor of 2 that was not correctly simplified, maybe a latex error), but I think that it should be more general and easier to prove that for any w the vector from w to its image f(w) is the orthogonal vector to the plane of the reflection.
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.
That's an absolutely obvious step though? As in, detailed lecture notes should maybe elaborate with a sentence, but in a lecture I would not put this on the slides but mention the core point and expect students at this level (who should have seen some amount of more theoretical LinAlg courses by then) to understand how to do the 1 line calculation.
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.
-- 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.
10 LB = LB' 0Q 0A 0A'
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.
Not exactly the same material but U. Michigan has their Robotics 101 course up as well: Computational Linear Algebra, also in Julia.
https://github.com/michiganrobotics/rob101/tree/main
A good resource is Gerard Sleijpen's course: https://webspace.science.uu.nl/~sleij101/Opgaven/NumLinAlg/
I guess the title would better be "Numerical Linear Algebra Class in Julia at TUM". I.e. the "TUM" in the title does not mean that there's some new "TUM" version of Julia, rather that the class is at the Technical University of Munich.
This is a nicely comprehensive course, but it looks like it is pretty fast paced, especially in the last few lectures (some of those later slides definitely aren't finished).
As a reference, it looks very useful.