A scalar is any mathematical object which only has size and no direction is my guess. But alas, from my limited understanding. A scalar can be any number, complex or real. (I guess scalars can be other things than numbers, but that is as far as my thinking goes at this point.)
I just have to start remapping my brain, from the previous years of learning how to do math like a compouter to starting thinking about things, and understading, and proving theroems.
Perhaps. If it is too confusing, then ignore all that we said and go on as you were told in class. [Just one thing to remember: almost everything you learn can be generalised like crazy to more and more general structures.]
A set of things that can be added to each other, and a set of "scalars" to multiply the things (vectors) by, such that your usual algebra rules of associativity-distributivity-commutativity hold.
@N3buchadnezzar A vector space is a structure: it's a set with two operations -- (vector) addition and scalar multiplication. Here the scalars come from an auxiliary field: commonly this field is just $\mathbb R$ (or $\mathbb C$), but it could be more general than that.
A field is anything with addition and multiplication defined, both additive and multiplicative identities (0 and 1 respectively), and everything other than 0 has a multiplicative inverse (a reciprocal 1/x). And obeys the familiar rules.
@anon's description is spot on; just remember that you are not really allowed to multiply two vectors in general. It's obvious when you think of concrete vectors, but if you are working with symbols (e.g., "v"), it's quite easy to lose track of such things.
But anyway, suppose you are editing a 3D movie in the distant future. You are working with three dimensions in any frame, and the entire thing is four dimensions.
It depends. For just R-vector spaces you don't need to know about fields in general, just R. If you want the general study of vector spaces it would make sense to first know about fields.
I can not even by strong liquour. I blame my lack of understanding on the fact I can not mix alcohol and abstract algebra. Then these things would make more sense.
And yeah, we had a bit of theory about sets. Not a whole lot though, just the basics.
But vector spaces has enormous richness even with real scalar field. When we go more general, we miss out on many things, e.g., you can't define inner products etc. I consider this a big handicap.
I think the more important thing is to grok whatever you are learning.
Why? Essentially, the difference between rationals/reals/complex numbers on one hand, and integers/matrices on the other is this: you can take reciprocals in the first case, but you cannot in the second case.
@N3buchadnezzar I just gave you two more examples of fields...
@N3buchadnezzar I don't understand what you're saying. You've blamed it -- I don't know what -- on previous years of learning like a computer, on not being able to mix alcohol and abstract algebra, on the clock, and finally poor ability to multitask.
the set of numbers of the form $a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}$ can be considered a vector space over the set of numbers of the form $a+b\sqrt{2}$ (with $a,b,c,d\in\mathbb{Q}$
I guess waking up tommorow these things will make more sense. It does not seem all that complicated now, just a new set of language to learn. And a new perspective to look at things.
You can define exponentials involving arbitrary reals in terms of limits. And the reals themselves, if you want sufficient rigor, with Cauchy sequences modulo null sequences....
I was reading the answer, and that was the only thing that sort of bummed me out. (perhaps more things, If I read through it again, even more carefully. )
If one are to use e^{$\log a$} and then a series expantion to give a propper definition to the reals. then one has to have a solid definiton of $\log a$
So if one were to "calculate" or define sqrt2 one would write the sum and to define log 2, one would use the fact that it is e^x = 2 and to e one would again use the sum, and to find x, one could do an approximation i guess. Or use an approximation for log a
Well I just feel all of these powerrules are laid out in such a way, that once you have proven one, the rest follows much easier. But to prove the first one is a *****
Use A to prove B, use B to prove C, use C to prove A.
log(exp(x)) = x = exp(log(x)) is pretty much the only given you need. proving exp(a+b)=exp(a)exp(b) with series expansion requires a binomial theorem and a re-ordering justification, if that's what you want to know about...
I don't understand the line of argument you're intending to pursue. You only have the inequality $|\sin{x}|\leq|x|$ (which is strict for $x \neq 0$, but this is what's to be proved)
Well, yes, $\lim\limits_{x \to 0} \dfrac{\sin{x}}{x} = 1$. But the question is about the converse: if $a_n$ is a sequence such that $\lim\limits_{n \to \infty} \dfrac{\sin{a_n}}{a_n} = 1$ must we then have $a_n \to 0$?
I cannot vote since I have not enough rep in that tag.
@AsafKaragila I would have to formulate my question much better to put it at MO.
Basically I just wanted to know if I did not miss some use of choice in my proof, but I tried to formulate the question in a way like "if you know anything interesting about this, please put it here".
If you view it minimalistic, my question can be understood as: Solve exercise E3 and E4 from Herrlich's book. I think this is way bellow MO level.
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Maybe I should have thought about the problem a little more before posting it, but as I mentioned, when I saw Martin Brandenburg's question, I've decided to post this, too. (When there's already one question on the site which is very close to this topic; I thought it's better to post mine question too - since people are already thinking about closely related things.)
*******
BTW it's quite probable that I won't answer to pings too promptly - I'm sitting in a room, supervising two students who have an exam (they're solving the test now); so I turned my sound off.
I notice some people here paste Mathematica output as LaTeX, e.g. here. I'm wondering how much of this they have automated, and how much is just Edit -> Copy As -> LaTeX with manual editing (as it produces non-copy-friendly output)
If any of you do this, and have some parts automated, could you tell me how you are doing it?
@Szabolcs I think you have to wait for robjohn or JM to show up, as they're the only ones here in chat using Mathematica on a serious level as far as I know.
@tb Well the paper is about prime and maximal ideals in commutative rings and also about Gelfand algebras. (Gelfand algebra is new notion to me, but I guess you mentioned that paper because C(X) is a Gelfand algebra.)
He uses Boolean prime ideal theorem, which is equivalent in ZF to UFT, IIRC.
> There is one and only one polynomial ($P(x)$) , with the maximum > degree of $n$, which satisfies $P(x_i)=f_i$ where $i=0, 1, 2, \dots , n$
To prove its uniqueness, I'd use prove by contradition. So I assumed $Q(x)$ as another polynomial of maximum degree $n$. Then for $i=0, 1, 2, \dots, n$:
So suppose you have two polynomials $P$ and $Q$, each of degree at most $n$, which interpolate your $n + 1$ data points. Then $P - Q$ is a polynomial of degree at most $n$ and at least $n + 1$ roots.
@MartinSleziak Proposition 2 of Banaschewski says: BPI is equivalent to the statement "every Gelfand algebra has a maximal ideal and this seems to confirm what you state in the paragraph starting with "I've tried to ..."
(I haven't thought deeply about it). The paper is certainly related but I don't think it answers anything immediately. Also, you might want to have a look into Johnstone's book on Stone spaces.
Yes. But that's a polynomial. In order for the polynomials to have any kind of reasonable algebraic properties, you want to consider it a polynomial. (you want to take sums, and products of polynomials)
Mother of all gods, I have two finish this in less than two hours, and I have no idea where to start. Anyone got a good idea about how to use Koenig's lemma about trees, or how to use propositional calculus' compactness theorem?
Basically got to do with the roots of unity. If $3$ does not divide $n$, then $\zeta^n$ and $\zeta^{2n}$ are (in some order) $\zeta$ and $\zeta^2$. ($\zeta$ is a complex cube root of unity.) So $1 + \zeta^n + \zeta^{2n} = 0$ which means ...
Finally: when $n$ is divisible by $3$, $1 + \zeta^n + \zeta^{2n} = 3$, so...
@AsafKaragila I remember using Konig's theorem to prove infinitary version of Ramsey theorem. It's basically the proof given as an exercise here and carried in more detail here. Are you serious to talking to freshmen about infinite trees?
I recommend adding a qualifier: "additive inverse".
@N3buchadnezzar You don't have to feel bad: in the context of functions, an inverse is commonly interpretted as a "function inverse". But that interpretation does not make sense in this case; that's all.
I noticed that i got an announcer badge 2 hours back in both the math.SE and physics.SE on two different questions
Further it shows that the question for which i got the badge in math.SE has been removed by me (the author)...if its removed i wonder how it got announcer badge ?
adding to it both these questions are atleast 4-5 months old
man those mathematica people were hostile due to their short sightness
don't they realize the reason their stackexchange was shutdown was due to lack of support, and understanding reasonings people don't support it might just help them..
Hi all. Gotta go to a meeting for work. These meetings take a lot of prep and may take all day. Sorry I've been away for a couple of days and will be for a bit more.
@Srivatsan: I have most of what I did for the polynomial fit on [0,1] written up. one detail to go. I will finish it and post it as an answer to the question which already has an accepted answer, I see.
@robjohn Not my question (didn't know that David's answer was accepted) =)
@robjohn By the way, do you know how the distance (of $x^n$ from the lower-deg polynomials) behaves asymptotically? Is it like $1/n$? Or inverse polynomial in $n$? Exponential?
High upvote is a reasonable indicator that the answer is good. I have two caveats in mind (assuming I asked the question): (a) If the difference in votes is small (e.g., 15 vs. 17), then I wouldn't go by votes that much. (b) If there is a highly upvoted answer I do not understand, then I wouldn't choose it. "Perhaps the answer gives an enlightening point of view, but it didn't work for me."
That speaks only for me: I am not sure if there's a general bias.
Hey! I'd appreciate some advice regarding week calculation. I'm wondering if the [-character is just a parenthesis, or if it's a mathematical sign? en.wikipedia.org/wiki/…
As the single day of the year in that week number. Does not comply with the ISO standard afaik.
My previous calculation broke down in the last week of 2011, since both the starting and the ending week of year 2012 is week 52. Didn't see that coming.
I'm not sure what the best thing to do in that situation is
you probably don't wanna call the ISO C libs for date manipulation if you're using .NET
> %G, %g, and %V yield values calculated from the week-based year defined by the ISO 8601 standard. In this system, weeks start on a Monday, and are numbered from 01, for the first week, up to 52 or 53, for the last week
it says 53 is okay according to ISO there though (man strftime)
> Week 1 is the first week where four or more days fall within the new year (or, synonymously, week 01 is: the first week of the year that contains a Thursday; or, the week that has 4 January in it). When three of fewer days of the first calendar week of the new year fall within that year, then the ISO 8601 week-based system counts those days as part of week 53 of the preceding year
