Wikipedia:Reference desk/Archives/Mathematics/2010 August 20

= August 20 =

How do you write this augmented matrix in reduced row echelon form?
$$ \begin{bmatrix} 1 & 2 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix}. $$

115.178.29.142 (talk) 03:25, 20 August 2010 (UTC)

Subtract twice the second row from the first row

$$ \begin{bmatrix} 1 & 0 &-2 &-3 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix}. $$

Subtract multiples of the third row from the other rows

$$ \begin{bmatrix} 1 & 0 &-2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}. $$

This matrix is in reduced row echelon form. Bo Jacoby (talk) 06:42, 20 August 2010 (UTC).
 * Do one more step: add twice the second row to the first row:

$$ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}. $$ Staecker (talk) 11:47, 20 August 2010 (UTC)

That doesn't work since the (2,2)th entry x 2 + (1,2)th entry isn't 0; it's 2. [ personal attack removed ] —Preceding unsigned comment added by 110.20.55.15 (talk) 12:00, 20 August 2010 (UTC)
 * Your correction of Staecker's error (who does know the basic math but made a momentary mistake) is welcome. Your personal attacks are not welcome. Please stop this. -- Meni Rosenfeld (talk) 13:08, 20 August 2010 (UTC)
 * Heh- yeah my mistake. That's what I get for adding answers first thing when I wake up. Thanks Meni- Staecker (talk) 13:10, 20 August 2010 (UTC)

Comprehensive Linear Algebra Book With Relations To Other Branches of Mathematics
I've always wanted to find "the" Linear algebra book. A book that would serve me for my entire math life. But I never could find it. Hence I ask the gurus at this reference desk.

My ideal linear algebra book would: (a) Cover all the fundamentals of linear algebra such as what's covered in Herstein's Topics in Algebra in a nice and elegant way (canonical forms, bilinear forms, hermitian matrices, you know advanced topics + the basics) (b) Be significantly more advanced than this and cover topics in linear algebra that pop up deeply in other branches of math (say tensor algebra) (c) Be a pure math textbook.

So is there such a book? I like the style of Herstein's topics in algebra so a book with this style would be desirable. My problem with Herstein's topics in algebra is: it's great and solid but I want a more advanced book. I like my book to give some calculations but also lots of nice theory in a elegant way. + the book should cover as many advanced topics as possible. This is important. It should be comprehensive. If a series of texts is required, that's fine. I don't mind if my book requires math background in general algebra, in fact I want a book like that. Thanks guys ... Oh and I forget to add: this book should be primarily on finite dimensional vector spaces. It's alright if other vector spaces are covered, but I want the emphasis on finite. Thanks guys again. .. —Preceding unsigned comment added by 114.72.248.27 (talk) 07:29, 20 August 2010 (UTC)

''Please answer my question? I request you as politely as I can. Why do people avoid me? What have I done? Please answer my question. I've made my question sound as polite as possible Thanks guys ...'' 114.72.244.249 (talk) 08:37, 20 August 2010 (UTC)
 * Presumably no one can think of any such book ATM. This is not a mathematical question, hence it's not possible to ponder it and figure out a solution; if no one here happens to have encountered a book meeting your rather high demands, you may well get no answer.—Emil J. 13:12, 20 August 2010 (UTC)
 * I can't think of any myself but you might find something at List of important publications in mathematics.--Salix (talk): 00:50, 21 August 2010 (UTC)

simultaneous equation
Find the values of $$k$$ for which the simultaneous equations:

$$(k+1)x-3y=0$$ $$ 2x-ky=0$$ have solutions other than x=0, y=0

--220.253.222.146 (talk) 09:05, 20 August 2010 (UTC)


 * Geometrically, what is going on here is that each equation represents a line through the origin (0,0). A solution to the simultaneous equations is the co-ordinates of a point at which the lines cross. The lines will not cross at any other point apart from (0,0) unless the two equations actually represent the same line, in which case any pair of co-ordinates that satisfies one equation will also satisfy the other. From the condition that the two equations represent the same line, you can get a quadratic equation in k which has two (real) roots. The same quadratic equation falls out if you take an algebraic approach and think in terms of matrices and determinants instead. Gandalf61 (talk) 09:41, 20 August 2010 (UTC)
 * Or simply solve for one variable in the one and then substitute into the other. $$k = -3, 2$$. —Preceding unsigned comment added by 203.97.79.114 (talk) 10:32, 20 August 2010 (UTC)


 * This looks like a homework question. Wikipedia Reference Desks do not do people's homework for them.  We are happy to point people in the direction of suitable information on Wikipedia so they can read and learn, but there is no benefit to you if we do your homework.   Dolphin  ( t ) 11:33, 20 August 2010 (UTC)

[ vulgarisms removed ]

For the record, the answer is to use matrixes. It's as imple as 1,2,3 ... [ personal attack removed ] Your system of equations is

k+1 -3

2   -k

and your goal is to determine when this matrix above kills something to 0. (like maps (x,y) to 0) But it does so when it's determinent is 0 and this is like when -k (k+1) = -6 or -k^2 - k + 6 = 0. But the quadratic has a solution = 1 +,- sqrt(1 + 24)/-2. So k = 3 or k = -2. I repeat

<<<<>>>>

Write that on your homework assignment with the advanced matrix stuff and 'all. Your teacher will be very impressed and you'll get an A+. If she's a hot lady, maybe she'll go a bit further than that for your own pleasure ... —Preceding unsigned comment added by 110.20.55.15 (talk) 12:01, 20 August 2010
 * Please stop disrupting this reference desk, or you will be reported and subsequently blocked. -- Meni Rosenfeld (talk) 13:13, 20 August 2010 (UTC)

I'm just speaking my voice. Maybe I was vulgar in the 80's but nowadays the words I used (I won't state them cause clearly people have problems with it) are common. If you get out more Meni Rosenfeld, you'll see that. And yes, the guy who asked the question is a ******* if he wants his homework done for him. But ******** are ******** and so we might as well give the ******* the answers. I CAN'T see what I've done wrong. I CAN see why other people *THINK* I've done wrong. For some odd reason society frowns down upon telling the truth ... But why shouldn't I voice my opinions? BTW, now that you've told me I'll stop, but all I'm saying is that you guys are all against me. First some dude called Tango removes my question for no particular reason. Then some annonymous guy tries to block me, and then you tell me to "sign my post". Well, I complied. But now people don't answer my questions anymore. I mean what do I need to do to get some service around here? Don't you guys get paid for answering my questions? I'll report you guys to Wikipedia for not doing your job right ... I mean I don't want to, but I just lost my temper. I'm ignored. And some guy called Gandalf insults me above. So everyone hates me. I think I should just leave and save you all the miseries. Goodbye. Block me, I don't really care. But I'm not posting here anymore until you guys start treating me with some respect. —Preceding unsigned comment added by 110.20.55.15 (talk) 13:28, 20 August 2010 (UTC)
 * You seem to misunderstand a lot of aspects of how the reference desk operates. One of these is that no one here gets paid for anything. Another is that we don't do homeworks, which you'd know if you read the prominent box on top of this page.—Emil J. 13:50, 20 August 2010 (UTC)
 * right: homework exists to help keep class distinctions in a country (United States) that has all too many social upstarts and "mobility". The only thing keeping any uppity declasse' from rising through the ranks of education is the assignment of homework that other kids with better-educated parents will have help in completing but which that kid will not.  I think it is the last barrier to total communist anarchy dictatorship of the proletariat, and any cheating, or attempts by students to coopt other classes' parents (such as the parents who would contribute to the refernece desk) must be smitten strongly, quickly, and decisively.  If someone can't ask their own parent this question, then maybe that person should accept his lot in life instead of trying to pull down the average in a higher class to which he, of rights, would never accede. 92.230.70.110 (talk) 21:23, 20 August 2010 (UTC)

Stop talking bullcrap 92.230.70.110. If you want to see bullshit in it's finest form look above LOL. —Preceding unsigned comment added by 110.20.11.94 (talk) 00:57, 21 August 2010 (UTC)

Yoplait Yogurt commercial
In a Yoplait Yogurt commercial, the development of a woman's bone strength is represented as a curve that starts at (0,0) and rises quickly before dropping at a slightly slower rate which decreases, and then approaching the X-axis. Whether this is accurate or not I don't know, particularly since the curve breaks like a bone when it reaches the point where a woman's bones would break easily. But I recall seeing a function like this, which for values of x less than 0, F(-x) = -F(x). What might this function be? Vchimpanzee ·  talk  ·  contributions  · 21:36, 20 August 2010 (UTC)
 * $$x e^{-x^2}$$ ? --Qwfp (talk) 21:56, 20 August 2010 (UTC)
 * Thanks. That might be it. Where could I find a graph of that, because Google has failed me. Vchimpanzee ·  talk  ·  contributions  · 17:59, 21 August 2010 (UTC)
 * http://www.wolframalpha.com/input/?i=x*e^%28-x^2%29. Note that this is just one example, there are infinitely many functions matching your description. -- Meni Rosenfeld (talk) 18:18, 21 August 2010 (UTC)
 * But I think it's the one I was looking for. Thanks. Vchimpanzee ·  talk  ·  contributions  · 20:21, 22 August 2010 (UTC)

Irreducible polynomials and the tower law
Hi all!

I'm trying to solve the following:

Let K be a field and c ∈ K. If m, n ∈ Z>0 are coprime, show that Xmn−c is irreducible if and only if both Xm − c and Xn − c are irreducible. (Hint: Use the Tower Law.)

Problem is, I don't really understand how you can use the tower law when you don't know something is irreducible: surely you require irreducibility of f(x) for K[x]/(f(x)) to actually be a field? In which case, you can't really say anything about the degree of the relevant field extensions before you know they're irreducible? And obviously you can't say they're irreducible until you've proved the result, at which point it's moot.

For example, if Kmn, Km and Kn represent K[x]/(xmn-c), and so on, respectively, and we assume Km, Kn are irreducible, then [Kn:K][Km:K]=[Kmn:Km][Km:K]=mn I believe, but we can't then say '[Kmn:K]=mn' because we don't know that's a field necessarily, do we? I'd appreciate a nudge in the right direction, as I've been stuck on this for a while and I think it's probably a very short argument, just one I'm not getting - I believe I've proved Xmn−c is irreducible implies both Xm − c and Xn − c are irreducible via the contrapositive, but I didn't use the tower law, or the fact that m, n are coprime: simply write Y=Xm (assuming Xn-c reducible), then Xmn - c = Yn - c is reducible. Am I doing something wrong here? Or just missing the obvious?

Thankyou very much for any responses, much appreciated indeed - 84.45.219.231 (talk) 22:52, 20 August 2010 (UTC)
 * Just let α be any root of Xmn-c and consider K[α]. This is definitely a field; it's isomorphic to K[x]/(f), where f is the minimal polynomial of α (which could in general be any irreducible factor of Xmn-c). Algebraist 23:02, 20 August 2010 (UTC)
 * Right, well I got the half of the proof which I was missing, so thankyou! (Something along the lines of 'm divides this, n divides this, they're coprime so mn divides this and it's less than or equal to mn, so equals mn'...) - Out of interest, does this argument also enable us to prove the direction I succeeded in proving without the tower law etc? I can't see how you could use it to prove '<=>', rather than just '<=', but I could be overlooking something. Either way, thanks for the help, I suppose I should probably press on with this fiasco which is teaching myself Galois theory, though I'm quite convinced I'm totally useless at it! Cheers :) 84.45.219.231 (talk) 00:09, 21 August 2010 (UTC)

If you can't do Galois theory what hope do you have to suceed in math? So to help you, take the following advice: Compute as many Galois groups as possible. That should aid your disorder. No need to thank me. —Preceding unsigned comment added by 110.20.11.94 (talk) 01:01, 21 August 2010 (UTC)
 * I'm not sure to what extent that's sarcastic, but if I intended to remain useless at Galois theory, I probably wouldn't be here trying to improve :) 84.45.219.231 (talk) 01:40, 21 August 2010 (UTC)
 * Please ignore the comments of 110.20.11.94, who unfortunately is now trolling this reference desk. I am sorry that you had to see this. -- Meni Rosenfeld (talk) 17:26, 21 August 2010 (UTC)
 * Much appreciated Meni, I shall do! Always glad there are dedicated 'refdesk-ees' such as yourself and Algebraist about to help. 84.45.219.231 (talk) 20:18, 21 August 2010 (UTC)