I also think so, but seeing the answer at the back of the book i thought i maybe mistaken, it has something to do with inverse proportionality as you see, so i tried several times anyway.
Hello @AkivaWeinberger! What do you think about my problem 'Several timbers of the length 2 m were cut into small logs of 50 sm in 4.5 hours; how long wolud it has taken if they were cut in 40 sm?' is it solvable?
all the books go out of their way to define lie groups in terms of manifolds, but then they immediately say all the """""important""""" lie groups are matrix lie groups anyway
Starting lines of thoughts: if you want to embed the universal cover of SL2(R), let's call that G, in M(nxn)(R) then taking the derivative gives a finite-dimensional representation of sl2(R) (Lie algebra of G = Lie algebra of SL2(R) = sl2(R)).
@SamuelYusim i guess an analogy is every smooth manifold can be embedded into $\R^{2n}$ and so in theory you can study manifolds just as submanifolds of euclidean space, but i think a lot of things become harder/not as obvious (e.g. what notions depend on your embedding and what notions are intrinsic to your manifold)
I think what we needed before to justify from $L - \epsilon <\displaystyle \lim_{k\to \infty}{a_{k}}^{1/k} < (L + \epsilon)$ that $\displaystyle \lim_{k\to \infty}{a_{k}}^{1/k} = L$ is that $\epsilon$ had to be arbitrarily fixed.
Fixing it arbitrarily it yields to $\displaystyle \lim_{k\to \infty}{a_{k}}^{1/k} = L$
So somehow I want to say that the $n$-dimensional representation of $\mathfrak{sl}_2(\Bbb R)$ extends to an embedding of $\text{SL}_2(\Bbb R)$ in $M_{n \times n}(\Bbb R)$ by exponentiating it, or something.
By composing with the covering $p : G \to \text{SL}_2(\Bbb R)$ you get another embedding of $\text{SL}_2(\Bbb R)$ in $M_{n \times n}(\Bbb R)$ which induce the same map on the Lie algebra, so these have to be the same embeddings. Aka, if $\iota : G \to M_{n \times n}(\Bbb R)$ was our original, $\iota = \iota_0 \circ p$ for some embedding $\iota_0 : \text{SL}_2(\Bbb R) \to M_{n \times n}(\Bbb R)$.
But... that can't be an embedding because $p$ squashes stuff - it's a nontrivial covering map
I'm working through the proof that $L^{p_2}(E) \subseteq L^{p_1}(E)$ when $1 \le p_1 < p_2 \le \infty$ and $E$ is of finite measure. In this proof, the author defines $p = p_2/p_1 > 1$ and claims that $f \in L^{p_2}(E)$ implies $f^{p_1} \in L^p(E)$. My only problem is, how is $f^{p_1}$ a well-defined real-valued function? If $f$ takes on negative values and if, e.g., $p=1.5$, then $f^{p_1}$ would be complex.
@SamuelYusim hm i don't really know lie groups all that well - so im not sure. i think it's also not clear what you mean by your question.. what properties about lie groups do you care about? personally one reason they're important is because they are important examples of manifolds so..
I mean, same here. I really just want some confirmation that yes, this is a thing people make actual use of, and not just a generalization for the sake of generalization
well - even if there didn't exist lie groups that are not matrix lie groups (which is false) - then what you would have is an equivalent definition - and often you'll find different (equivalent) definitions being useful in different settings!
@SamuelYusim Off the top of my head, I think one of the unifying stories for me has been the geometry of Lie groups. Every compact connected Lie group $G$ has a Riemannian metric (you know what those are, right? Otherwise read it just as "metric") that's compatible with the group structure (read: $d(x, y) = d(gx, gy)$). Under that, you get a geometric theory in which the geodesics (locally distance minimizing paths) of $G$ are the 1-parameter subgroups, i.e., homomorphisms $\Bbb R \to G$.
Question: How does bi-invariance of the length metric interact with the bi-invariance of the Riemannian metric? Does the former imply the latter? I'd think so.
@SamuelYusim actually an example where i think the advantage of knowing things about lie groups is clear is for example if you want to prove that $U(n+1) \rightarrow U(n+1)/U(n) \isom S^{2n+1}$ is a fibre bundle (which is an important bundle in topology!) - in the sense that I think trying to prove this knowing what the group is confuses things imo, but where general facts about actions of lie groups give you the result.
I have a space $X$ such that for every point $x \in X$, $x$ lives in a linear subspace $S \subset X$ of vector space dimension $n$; however, it's possible for two different points to live in subspaces of different dimension Does this make $X$ a topological manifold? I don't think so, since the definitions I've come across seem to require a "global", "uniform" dimension. If not, is there a name for such a space?
I have a question regarding the notation of a linear algebra question regarding linear Transformation.
Lets say there is a linear operator on $R^2$, Let $L$ be the operator defined by $L(x) = 3x$. Can another way of writing this be $L(x) = 3x_1$? So that
The wedge sum and the free product of two topological spaces would have the same fundamental group by Van-Kampen, right? Because one is $\pi_1(X)*\pi_1(Y)$ and one is $pi_1(X) \substackl{\{\text{point}\}}{*} \pi_1(Y) = \pi_1(X) * \pi_1(Y)$?
@TedShifrin One of my primary complaints about this school is that very little attention is paid to the teaching of undergrads. It might be worse than berkeley in that regard.
Yeah, that's why they should have a standard text and syllabus.
This is always a confusion, @Jon. People write $v_1$ and $v_2$ for two different vectors sometimes, but they also write $v=(v_1,v_2)$ and mean those are the coordinates. You have to have context.
My complaint (as associate dept head and chair of curriculum before that) was faculty who ignored the departmental syllabi in the name of "academic freedom." Really fair to students, that is.
right, I thought it was okay because it's not an exam on fundamental groups or van kampen's theorem, that was last semester, but Antonios is saying that it's related enough to be potentially dishonest, and I see where he's coming from
You're well aware of this after your experience this semester, but it's a matter of what their background is, what the goals of the course are, and what it's supposed to prepare them to do.
Right, @Jon. If your definition you're supposed to check says $L(v_1+v_2) = L(v_1)+L(v_2)$, there the $v_1$ and $v_2$ are two different vectors. Nothing to do with coordinates. This is the confusion I was talking about before.
@TedShifrin So, that class was 2 and a half hours a day, every weekday. We had a break, I imagine some number of people got on their phones, though I don't think Laci especially cared. At least it was a chance to stretch
The course description "Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer's rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms."
Plus exercises ... For linear algebra, it takes time to make up problems where the numbers aren't horrid. (Feel free to steal more from me if you want, @Antonios. :))
The AoPS linear algebra I'm teaching drives me nuts. They make no efforts to avoid horrendous arithmetic. Why not make the numbers in lecture come out reasonable?
This class was the first one that got my into math actually, I had a graduate student instructor who explained it all in terms of geometry and it really made a lot of sense to me.
It was also the first time I did math that wasn't calc.
@Antonios: They couldn't consider my book because we didn't have enough differential equations stuff in there for the Berkeley curriculum. They didn't ask if we'd modify it ... But I think our book was too proof-oriented.
Yeah honestly linear algebra was also what got me into the subject as well, so I'd definitely put a lot of emphasis on making it a more interesting/inspiring class
I think students should learn to do row reduction (with reasonable numbers), but then feel free to use technology after that (except exams). The issue is to understand the concepts enough to know how to solve problems, not do mindless arithmetic.
Just remember the majority of your students aren't little yous or Demonarks.
They're not going anywhere near advanced math or grad school.
yeah, it's sort of frustrating. Because of course I feel like making the course I would have wanted when I was in their position, but it's not what everyone wants.
Yeah the tricky part is when the class is designed to be a service class of some sort, because then the actual practice of it may be kinda boring but at the same time, there will be complaints if you try to teach the class in a way that's more fun but doesn't get to the content that people need to see
That's why I always did plenty of applications even while I was trying to teach them to learn to write proofs. But that requires office hours to work with the students.
But I think there's a standard level of work where you can just say look, if you're not willing to put this much effort then take the crap grade and deal with it
@Jon I don't claim to be good at advanced linear algebra, but I think the most insightful thing for me in learning the subject is reading 3 different books bits at a time.
There are a few different perspectives you want to understand before you really start to see the bigger picture.
Honestly I might at some point learn some computer algorithms for matrix factorization, that could be an interesting balance of both being somewhat interesting and useful to know
That's in numerical analysis type courses, Demonark, not beginning linear algebra. Plenty of students are interested, but not in an introductory course.
Hmm, if neither that nor the more geometric and/or algebraic content is there then I'm not really sure how to avoid making LA "the dreaded class" (before college I remember being told by some folk that it was one of those boring classes you just had to hold your nose and power through)
So, my first touch with LA was in the REU class I mentioned above. It started off with graph theory for a bit, then it did LA for about 3 weeks, then some stuff in the intersection. We didn't really emphasize things especially geometrically? Like we were emphasizing how you could go back and forth between linear maps and matrices, but a lot of our earlier proofs didn't necessarily restrict to R, we only talked about inner products much later, that sorta thing
I kind of appreciated that somewhat. Like, there was some slight chaos at the beginning because we our professor didn't come until halfway through second week, so first week was graph theory with one of his grad students, and first two days of second week was with someone else
@Ted: Do you know where I should look to see formulas for $\Delta_A([\sigma, \psi])$ where $A$ is a connection and $\sigma, \psi$ are sections of a bundle of Lie algebras? I would guess that the usual slogan "products of harmonic functions are not usually harmonic" is true, but would like to understand it well.
But the two people had different working definitions. Our first guy defined the rank of a linear map to be the dimension of the image while our second guy defined the rank of a set of vectors to be the max number of linearly independent vectors
Should respect the product structure in some sense. I have a slightly more special case in mind, where $A$ is a connection on a $G$-bundle $E$, and there is an induced connection on $\text{End}(E)$.
