Lets say I have an $m+n$ dimensioned vector space $V$. Let $W$ be a subspace of dimension $n$, spanned by $\{d_1,d_2 \cdots d_n\}$. Let the extended bases for $V$ be $B=\{d_1,d_2 \cdots d_n, e_1, e_2 \cdots e_m \}$. Let me define $V_j$ to be the $m+n-1$ dimensioned subspace of $V$ spanned by $B_j=B \setminus \{e_j\}$. Is $V_1 \cap V_2 \cdots \cap V_m=\text{span}{\{d_1,d_2 \cdots d_n \}}=W$? Just asking so that I am not hallucinating any details
for that exercise he asks me to calculate the work of $\omega$ along the curve $\gamma (t) = (\cos(2t), \sin(2t)) , t\in[0,\pi]$, but then $\int_0^{\pi} [e^{\cos(2t)} (\sin(\cos(2t) + \sin(2t)) + \cos(\cos(2t) + \sin(2t)) \cdot \\ (-2\sin(2t)) \\ + e^{\cos(2t)} \cos(\cos(2t) + \sin(2t)) \cdot 2\cos(2t)] dt$
@PM2Ring That projection will take my equilateral triangle on the sphere and flatten it, but the result won't be a Euclidean equilateral triangle like I'm hoping to get. If I'm not mistaken, it'll take the 72° interior angles and make them even wider.
remember that given $\omega=\sum_{i=1}^n \omega_i(x)dx_i$, you can consider the vector field $F=(\omega_1,\dots, \omega_n)$, $\omega$ is exact is equivalent to $F$ is conservative
in $\mathbb R^3$, $\omega$ closed if $\operatorname{rot} F=0$
@SophieSwett Oh. I'm not sure what the Lambert does to the angles. A gnomonic projection can give you a 60° triangle because it maps great circles to straight lines. But as you can see, it definitely won't preserve area.
we have only 2 linear algebra courses in our entire 5 years of study, and the first one is a computation based course open for all science students, the 2nd one uses Sheldon Axler. But looking at hoffman kunze, i cant help but notice a decent chunk of stuff in there is left out, be it entire topics or in-between theorems. Is this how it generally is?
@SoumikMukherjee entire chapter-4 of hoffman is left out (polynomials), grassman ring, multilinear functions, modules, commutative rings (hoffman develops determinants using rings I believe), and no bilinear forms. much of 9th chapter of H&K deals with forms which is not there I guess
The supremum axiom, so far I unde stand, characterizes all subsets of R with a supremum. Is there some equivalent statement which does the same for Q? That is, what are the necessary and sufficient conditions such that the supremum exists for a subset of Q?
@CatharticEncephalopathy asking for sufficient and necessary conditions for X too broad of a question. Sure, it's equivalent to X. But what exactly do you expect the answer to be?
What form of answer do you seek ? Many answers would fit your question, but what form of conditions do you expect ? — Maxime7 hours ago
@SineoftheTime Are you there by any chance for a moment?
$y'' - 2y' + y = \frac{e^x}{x+2}$ , I found that $\lambda = y$ it is a solution of $ay^2 + by + c = 0$ of multiplicity $m$. So I can use $x^m \cdot e^{\lambda x} \cdot (A_0 + A_1x,...)$ but since I have a rational fraction, I can't use a standard polynomial expression with $A_0$ etc, can I use as a particular solution $\frac{x^2e^x}{x+2}$?
I used $m=2$ since the root $r=1$ of the characteristic equation is double
@nickbros123 You will probably learn about polynomials and rings in some other courses. But the LA course should contain multilinear functions and bilinear forms
Let $X, Y$ be metric spaces. Consider the product set $X×Y$ with the product metric (defined below). Show that any set of the form $B(x,r) × B(y , s)\subseteq X×Y$ is open in $X×Y.$
Let $(X, d)$ and $(Y, d)$ be metric spaces. The function $$d((x_l,y_1), (x_2,y_2))=\min \{d(x_1, x_2),d(x_2,y_2)\}...
there are typos all over that question and it is too long, as usual. the question is whether that formula, or that formula redefined so that it involves y_1 in an obvious way, defines a metric. it doesnt.
maybe try "max" instead of "min." try stating the source of the problem so people can check which errors are yours and which errors are in the source.
horrible notational practice to use the same letter, "d", for three different metrics, as you seem to acknowledge by changing the name of the metric on X x Y. if that is the source's fault and not yours i would mention it. a lot of people will skip right by a question like this if they can't tell whether it is a question about an unreliable source (like a textbook that has a typo in it), or a question from an unreliable source (i.e. an OP who makes typos a lot)
[the putative typo in the formula defining your putative metric on the product space is that the second thing in the 'max' should probably be d(y_1, y_2). in the current context, there is no reason why "d(x_2, y_2)" would have any meaning because it purports to compute "d" between elements of different spaces]
i now click into the history of this question and note that "max" became "min" in an edit. if it was originally "max" it should have stayed "max," unless, again, this is something in a source you are drawing from, and not your own creation (in which case i would mention that)
the "d(x_2, y_2)" issue is maybe the outstanding issue. is that a typo in your source, or is it your typo? please note that rigorous type checking of the problem statement would reveal that we do not have a given way of computing "d" between an element of X and an element of Y
as for the proof you might have noticed already that $B((x, y), r) = B(x, r)\times B(y, r)$. Moreoever if $(t, z)\in B(x, r)\times B(y, s)$ then there exists $r_0$ and $s_0$ with $B(t, r_0)\subseteq B(x, r)$ and $B(z, s_0)\subseteq B(y, s)$. By replacing $r_0$ and $s_0$ with $\min(r_0, s_0)$ we can assume $r_0 = s_0$. Thus $B((t, z), r_0)\subseteq B(x, r)\times B(y, s)$
@SoumikMukherjee and @leslietownes With your suggestions, I checked the errata of the book I was reading and yes, it's a typo... it should've been max instead of min and with this modification, I was able to solve the problem... the moral shall always be: When stydying a book, if you get stuck before wasting time, check the errata...
This seems so frustrating when so much time gets wasted just because of a typo...but ig it can't be helped.
Thanks to both of you, for pointing it out! I'll now delete my original post as it's useless now.
the more you study a subject, the easier it becomes to spot 'obvious' typos, until it wraps around and you no longer notice 'obvious' typos because you mentally fix them every time you read them. :)
there's a similar thing with human detection of typos and spelling errors, you'll be very bad/inefficient at it if you don't know the language, you'll get better, then eventually you learn the language too well to do it as well as someone who doesn't know the language as well
we are jsut challenging each other who can simplify the 2 terms we foudn in one still respecting the rules
but we both got stuck
so here we are
if you are worried this is an exam you can with until tomorrow to prove that this is not tomorrow please don't take too long or my friend is gonna keep asking em if I found it
if you are still worried about the issue here is the solution I found
when MO started they were very enthusiastic about encouraging people to use their real names, and i remember thinking, welp, there are definitely two ways it can go if people in your department find out how much time you spend in a place like this
sine: well, sort of like user said. for older / established faculty it would probably not make any difference. if you were on your way up in the academic world, e.g. not in a tenured position, or really in any position where you might be looking for another job sooner rather than later, being seen as spending a ton of time online might not be professionally beneficial, to put it mildly.
and yeah, using real names and institutional affiliations anywhere, even outside of academia, maybe only seems like more of a good or neutral idea if you have never been harassed online based on your perceived identity. most of the people who push that uncritically and unreservedly tend to be men, for example, and i don't think that's a coincidence
ephe it seems like something nlab would have. or if this is something where there are like 20,000 papers about what the "right" definition of bicategory is, maybe nobody would have
i don't know enough about the theory to tell which one
The supremum axiom, so far I unde stand, characterizes all subsets of R with a supremum. Is there some equivalent statement which does the same for Q? That is, what are the necessary and sufficient conditions such that the supremum exists for a subset of Q?
