yeah, I think from the perspective of linear algebra alone, the behavior of countably and uncountably infinite-dimensional vector spaces isn't all that different
Can someone who really enjoys analysis or topology (or something like that or any subfield, in particular not number theory nor algebra, so mostly the study of continuous objects) tell me what made you start enjoying it in the first place?
I actually loved analysis, even though I didn't much like my first professor, because the whole idea of estimates appeals to me and is very visual. Algebra tends often to be taught as symbolic manipulation (although books like Artin's are fabulous and unify concepts beautifully).
I liked differential topology, and then differential geometry after that, because of the power of the inverse function theorem and the way one can use intersection and degree theory, very visual things, to prove big theorems that otherwise require algebraic topology machinery that has very little intuition to its algebra.
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In my long teaching career, I successfully hooked a large number of students on mathematics with the first year course (baby analysis, but with all the applications in standard calculus as well) taught out of Spivak's Calculus. And, surprisingly, most of the students whom I taught who went on to do a PhD in ended up in geometry of some sort. I never expected that.
It's interesting. Some people stop liking math when they have to deal with concepts and not memorized formulas. Flip side: some people (like the hordes of kids in my Spivak calculus class who had always been good at math but didn't really like it) get excited when they finally get to proofs/concepts and can see interesting things going on.
It seems implausible to me that some people stop liking math when they have to move beyond memorizing formulas. It seems more plausible to me that those people tolerated math because it was relatively easy to them and gained a distaste when asked to understand the things they were doing because they didn't have much interest to begin with.
This seems to be the case with the "will this be on the test?" crowd
@MikeM: In my 40 years of advising and teaching freshmen, I experienced a lot of students who had always been good at (formulaic, programmed) math in high school and wanted to be math majors. That typically lasted one semester. The flip side was the people who had never considered math as a major (because they were so bored in high school) who then get enraptured.
On the other hand, in my experience, it takes a deft touch. Some people who try to subject the kids in that Spivak course to utter formality and 150% rigor drive the kids away very quickly.
That course, sadly, has gone by the boards. AP BC exam took care of it, finally. So my multivariable math course became the magnet (or destroyer).
I often was very explicit and told students (and put stars by it on the board) when highly important things were guaranteed to be on the test. Some of them never learned to trust me.
Yes, I think those people just enjoy running algorithms.
I get where they're coming from. There were these "puzzles" I did when I was a kid where you were given 64 little 5x5 grids of squares which are jumbled on the page, and supposed to pencil in the corresponding squares into the correct spot on a big 40x40 grid, and in the end you'd see some cartoon character. I liked running that algorithm.
@TedShifrin I think I agree with your "thorough isn't always good"; my current algebra course is actually a 3rd semester undergrad course and 90% of it has been category theory and homological algebra (not just for modules, which I'd guess is more common in a first course on homological algebra, but in general abelian categories)l I understand now why Lukas is such an algebra machine lol
Some rigor just isn't worth the pain. I never presented all the proof of the Change of Variables Theorem, for example. Nor did I take the half-hour to prove that mixed partials are equal. Both of these proofs are in the book and the "interested" student can read them. I think I proved pretty much everything else, but I tried to provide understanding/motivation more than rigor for the sake just of rigor.
Speaking of Lukas, @Edward, have you seen him or heard from him?
I didn't like linear algebra much when I was in high school. It was mostly just computations that I didn't find enjoyable and I never managed to memorize how to multiply matrices. Then in my first semester of uni, I was blessed with a fantastic algebra prof who opened my eyes for the beauty of subject - both the conceptual elegance of the abstract side and its correspondence to the explicit geometry and calculations on the computational side. That's when my inner dirty algebraist awakened.
I felt like I just awakened my third eye when I learned that matrix multiplication is the way it is, because it corresponds to the composition of linear maps
@Thor: A serious comment. This shows why the right definition of multiplication * vector is as a linear combination of columns, but everyone always does dot products with rows because they only care about systems of linear equations.
a symmetry of that field would be an isomorphism to itself . And we observe that these isomoprhisms send roots to other roots of the polynomial. So all these symmetries have the structure of a group . then i can relate the solvability of the group to the radical extensions of the splitting field. My question is that solvability between the group and the field extensions is not an obvious one to someone now having seen modern algebra.What did Galois observed ?
im just trying to see how the symmetries of the field that contains the roots of polynomials gives me a tell about writing those elements is a specific form.
yeah, I also like to think of conjugation that way and it's also how it usually appears in more explicit contexts
I don't know a good way of relating that interpretation to a good motivation of normal subgroups as subobjects to quotient by, though
@Manolis the rought point is that being able to write the roots of a polynomial in terms of its coefficients via addition, multiplication and taking roots corresponds to constructing a tower of abelian extensions (omitting some technical details such as passing to Galois closures), which corresponds to an abelian subnormal series of the Galois group, which is solubility by definition
the least trivial step in all of this is the realization that any cyclic extension of degree n (in characteristic not dividing n) comes from adjoining an n-th root
if your ground field contains n-th roots of unity*
I think the proper way of motivating the proof of that result comes from Galois cohomology, but I don't really know that story
if you want the history, take a look in Stewart's Galois theory or even in Kleiner's A History of Abstract Algebra
like X^3-2=0 consider the 3 roots a=cube root of 2 , v= a complex root. Now an isomorphism exist of Q(a,v)= Q(a,v^2) and Q(av,v^2)=Q(a,v) and if i check all the symmetries of Q(a,v) sending a and v to another root all these isomorphism (symmetries) form S3=D3. Now Dn groups are solvable. So beeing able to get the roots through reflections and rotations translates exactly to constructing numbers through + x and n-roots
no thats wrong its like im saying cube root of 2 can be constructed with ruller and compass
actually on ruler compass construction im allowed only square roots and in the example i said n-roots maybe its not that wrong what i said
Guys I have just "met" this question https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb/44129?newreg=873d964b774b4018927d6c1551aaad72
I'd like to understand the answer. I think I understand half of it - which is great.
When the answer starts talking about Tarski, I'm completely lost
What should I read to become familiar with the topic?
