Recent content by mathwonk

Those notes 844-1, 844-2, of mine linked in post #12, have as prerecquisite the earlier notes 843-1, 843-2, on the same webpage: https://www.math.uga.edu/directory/people/roy-smith The notes 843-1 are on groups, the notes 843-2 use groups and specifically Galois groups to give a necessary...

apparently (by Smale's h-cobordism theorem) the crux is the pseudo-isotopy class of the (orientation preserving) gluing map: https://math.stackexchange.com/questions/1807683/what-is-the-relationship-between-diffeomorphisms-of-the-sphere-modulo-isotopy-an apparently the equivalence classes of...

some arithmetic: according to AI, there are about 50 million public school students in the US, and to extend the school year by 10 weeks (2 week summer vacation instead of 12 weeks) would cost almost $4,000 each, or almost 200 billion extra dollars. that's assuming the teachers and staff are...

my wife came back from the fancy store- with- everything today with a bag of polenta, and a photo of the bag of "gluten free polenta" next to it, priced at a dollar more. then she reminded me that all polenta is gluten free, being corn. The two bags had the same brand name and same ingredient...

I am not an expert but here are some suggestions from math stack exchange: https://math.stackexchange.com/questions/153022/a-good-reference-to-begin-analytic-number-theory

This reminds me of how puzzled I was, when writing my first algebra book, by this question: find a vector basis of the product of a countable number of copies of the rational (or real) numbers. I.e. consider the space V of all sequences of rational numbers. This is much simpler than the space...

Perhaps the answer is contained in your first point. I.e. perhaps one should teach sequences first and thoroughly, and only treat series later. This is done in Richard Courant's excellent calculus book vol.I, where sequences are given importance from page 27, and, after Taylor series, general...

fresh, I suspect you are helping train a bot.

In post #28, what matters is that the vectors xj span. And that they are eigenvectors is irrelevant. Indeed the eigenvectors of a matrix do not always span. (Eg. a 3x3 matrix with all zeroes except 1's just above the diagonal, and a 3x3 matrix with all zeroes except 1's everywhere above the...

Sciencemaster:..."Is there anything else I'm missing here?" I would say you are missing the basic property of a linear transformation, namely it is entirely determined by its effect on a basis. For the same reason, a linear transformation only has one matrix in the standard basis. done...

k is the field of scalars in the matrix, e.g. k could be the reals. I think your analysis as to canceling out makes sense, but to me it is backwards to start from the non unique diagonal matrix and argue that the original matrix is unique. That's why I gave an argument for the uniqueness of...

I took a quick look at that book and wondered why you were interested in such a tedious approach to topology. Then I noticed some very nice theorems on pages 54-55 with simple characterizations of a circle and an interval. E.g. apparently any compact connected metric space which is...

Thanks to pasmith and fresh_42, I think I finally understand your question. To paraphrase: A linear transformation is entirely determined by what it does to any basis, hence if a linear transformation has enough eigenvectors to form a basis, then those eigenvectors and their eigenvalues do...

Relax. You are not bad at math, rather you are clearly better than average, way better. But you seem to be blinded by something I shared with you at your age. You have been conditioned, as I was also, to care more about your scores relative to other people, and being told you are a genius...

I believe all 6 similarities are possible with the numbers as given. E.g. with the numbers as given, one can apparently have angles BAE, ABE, ADC, all equal. Then the "rotation" taking vertices A-->D, B-->A, E-->C, seems to be a similarity from triangle ABE, to triangle DAC. Does that seem right?