Recent content by mathwonk

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?

I admit I still find this confusing, but it seems to me that all 6 possible similarities can occur, still with only two final answers of course. I mention this because it seems to me the question can be viewed as having 2 parts: 1) how many different ways can the triangles be similar, and then...

yes, I believe that is what I said.

the triangles could be isosceles, and hence have self similarities. right? i.e. we seem to be making the unstated assumption that the similarity must fix angle A. It seems angle BAE could equal angle BEA, no?

What surprised me was that there seem to be more than 2 ways the triangles can be similar, but they yield only 2 results for the question asked. Did this surprise anyone else? Or am I wrong?

I am also a retired college math teacher (over 40 years teaching). So I am on the side of Mark44 in this matter. I do not disagree with anything anyone else has said. I just know from decades of experience that it is hard enough to teach even the most basic facts, that one does not ever have...