I'm working on Fulton Algebraic Curves exercise 2-14. I'm stuck on something that I think should be simple but I'm not seeing. The problem is as follows: A set $V \subseteq \mathbb{A}^n$ is called a linear subvariety if $V = \mathcal{V}(f_1,\dots,f_r)$ for $f_i$ of degree $1$. Show that if $T$ is an affine change of coordinates, then $V^T:= T^{-1}(V)$ is also a linear subvariety. From the definitions we have that $$V^T = T^{-1}(V) = \mathcal{V}(\mathcal{I}(V)^T) = \mathcal{V}(\mathcal{I}(\mathcal{V}(f_1,\dots,f_r))^T) = \mathcal{V}(\mathrm{rad}(f_1,\dots,f_r))^T)$$ I would like to say that this is then equal to $\mathcal{V}((f_1,\dots,f_r)^T) = \mathcal{V}(f_1\circ T,\dots, f_r\circ T)$ which is then evidently a linear subvariety since $T$ and each $f_i$ are degree $1$ polynomials, so the compositions $f_i\circ T$ are also degree $1$. But I don't see why the ideal $(f_1,\dots, f_r)$ is necessarily radical. Any help would be appreciated.
Let $K$ be any field. Is it true that:   If $f_1,\dots, f_k \in K[x_1,\dots,x_n]$ are polynomials of degree $1$, then $I=(f_1,\dots, f_k)$ is all $K[x_1,\dots,x_n]$, or, if it is a proper ideal, then is it prime? In particular, is it true that $K[x_1,\dots,x_n]/I$ is isomorphic to a polynomial ring (in the case that $I$ is proper and hence prime by the statement above)?   If they are true, how can i prove these statements in an elementary but rigorous way?  P.S.: Birth of my question: I found on my book that "affine varieties given by linear equations are irreducibile", so I translated (I don't know if I have made it right) this geometric statement into the analogous algebraic version.
For $x,y\in\mathbb{N}$ how many ordered pairs $\left(x,y\right)$ satisfy   $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{pq}$ where $p,q$ are distinct primes? Allow questions to be saved as drafts prior to posting Integral equation problem How to link data to a polygon ArcMap 10.3 I have a map with different polygons (118). Each of this polygons represent diferent vegetation types.  For each of those polygons I have data of which species there is and how abundant they are (the abundance is measured with a scale from 1 to 3).  I would like to "link" (I am sorry if its not the correct way of name it) the data of which species and abundance there is in each of those polygons.   I would like afterwards to be able of selecting the different polygons containing the same abundance and species.  In the polygons I created there is no atributes associated other than (FID, shape and Id. the 3 of them created alone by the program when I drew them) and the tables I have in excel with the data I want to "link" to the polygons contain a column called species and another abundance.  As you can see, I'm pretty new using the program...    As I have tried to do what you all suggest in the comments down here, and I couldn't achieve what I was wanting to do (provably because of my bad explanation before...)  I leave a picture:  As you can see in the attribute table, that is how every attribute table of each polygon (which represents one area) looks like. On the right there is how I have my data in excel. Could you please suggest something I could do in order to add to the polygons attribute table the data belonging of species and its abundances (polygon number 1 has to have all the species (Acer mons, Acer pseu, Acer repens….) and its abundance (which are the numbers of the row Area1). What's the fastest connection speed / shortest elapsed time using Migration Assistant? When using Migration Assistant to move users, files, and folders from an old Mac to a new one, you have a few different choices, depending on the Macs involved. Presuming that both Macs have appropriate ports, which ports should be used in a real-world situation? Please do not provide quotations of theoretical port speeds; I'm looking for real-world experiences as to what will get the job done faster.  Port choices:   FireWire 400 in target disk mode FireWire 400 to 800 in target disk mode FireWire 800 in target disk mode 100 Megabit Ethernet to Gigabit Ethernet in the OS Gigabit Ethernet to Gigabit Ethernet in the OS Thunderbolt cable in target disk mode I have studied from Griffiths, the relativistic form of momentum is $$p = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} m_0v$$   Now when I evaluate the momentum for photon, I just insert $v=c$ and $m_0=0$ and I get $p= 0/0$. How does it make sense?   Can you tell me that where I am wrong? Why doesn't the Windows command prompt window maximize to the full screen size? Anyone know why doesn't the Windows command prompt window maximize to the full screen size?  To the best of my knowledge, even in the new Windows 7 the command prompt window(cmd.exe) doesn't maximize to the full screen size like all other windows. This behavior is quite annoying for me, and I'm sure to many others as well.   Note: The size of the command prompt window can be , but the size remains fixed regardless of the screen size. I have a list of items   John ID Matt ID John ID Scott ID Matt ID John ID Lucas ID   I want to shove them back into a list like so which also means I want to sort by the highest number of duplicates.   John ID 3 Matt ID 2 Scott ID 1 Lucas ID 1   Let me know how I can do this with LINQ and C#.  Thanks All  EDIT 2 Showing Code:      List&lt;game&gt; inventory = new List&lt;game&gt;();     drinkingforDataContext db = new drinkingforDataContext();     foreach (string item in tbTitle.Text.Split(' '))     {          List&lt;game&gt; getItems = (from dfg in db.drinkingfor_Games                                where dfg.game_Name.Contains(tbTitle.Text)                                select new game                                {                                    gameName = dfg.game_Name,                                    gameID = Boomers.Utilities.Guids.Encoder.EncodeURLs(dfg.uid)                                }).ToList&lt;game&gt;();          for (int i = 0; i &lt; getItems.Count(); i++)         {             inventory.Add(getItems[i]);         }     }      var items = (from xx in inventory                  group xx by xx into g                  let count = g.Count()                  orderby count descending                  select new                     {                         Count = count,                         gameName = g.Key.gameName,                         gameID = g.Key.gameID                     });      lvRelatedGames.DataSource = items;     lvRelatedGames.DataBind();   This query displays these results:   1 hello world times 1 hello world times 1 Hello World. 1 hello world times 1 hello world times 1 hello world times 1 Hello World. 1 hello world times   It gives me the count and name, but it doesn't give me the ID of the game....  It should display:   6 hello world times   234234 2 Hello World.  23432432 How do I run two isolated instances of firefox? For testing purposes, I want to install two different instances of firefox on ubuntu - so that they don't share cookies. (I want to simultaneously login with both instances as different users to our website).  How do I do that? I do it with Firefox &amp; Opera, but I prefer Firefox (because of firebug). Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$ What program can be used to package software into one executable? I need to package an application into a single executable file (so that I don't have to install it, so that it will be portable).  Does anybody know what I can use to make a program portable? Why was a moved question deleted? Moderator to whom deletion is attributed claims they didn't do it A  appears to have been deleted, and deletion is attributed to a moderator,  deleted by Sathyajith Bhat♦ 12 hours ago , however,  (however, they did previously (unsuccessfully) tried migrating it to another site in the network where they're also a moderator, but migration was rejected by another moderator, it was then on hold on webapps, and then the hold was removed, which was supposed to have been the end of it).  Can someone explain what's going on here?  And restore the question that clearly wasn't meant to have been deleted, was a HNQ, and has 75 votes on the question, plus 74 votes on the top answer, and 56 on second answer; and 19 on third?  There's a lot of links to this question, too, and now they're all broken, making SE look like a community hostile to a popular question ontopic for the site it was on. What is the mass density distribution of an electron? How do I repair a corrupted Firefox places.sqlite database? I had some problems with my RAM (bluescreen several times, Windows XP) and now are my Firefox databases damaged. Firefox is working, but my history is gone and it's reporting several inconsistencies and errors when executing pragma integrity_check on places.sqlite:     database disk image is malformed   Now the question, how do I repair SQLite-Databases? If $\cos x=\cos y$ and $\sin x=\sin y$ is $x=y$ or $x=2\pi,y=0$ or $x=0,y=2\pi$? $x,y\in[0,2\pi]$ I know that if $\sin x=\sin y$ then $x=n\pi+(-1)^ny$  and $\cos x=\cos y$ implies that $x=y+2k\pi$ or $x=-y+2l\pi$. Since the latter does not fit the form above $x=y+2k\pi$ which implies that $k=0,1,-1$. Hence, we have x=y or $x=2\pi,y=0$ or $x=0,y=2\pi$. Why does the image not appear? An easy bash completion tutorial? What does: Error Broken count >0 mean? I have the message:   Error Broken count &gt;0 Check if you are using third party repositories. If so disable them, since they are a common source of problems. Furthermore run the following command in a Terminal: apt-get install -f Details: The following packages have unmet dependencies: firefox-globalmenu: Depends: firefox (= 19.0+build1-0ubuntu0.12.04.2) but 19.0+build1-0ubuntu0.12.04.1 is installed   Any suggestions? Calling PHP functions within HEREDOC strings In PHP, the HEREDOC string declarations are really useful for outputting a block of html.  You can have it parse in variables just by prefixing them with $, but for more complicated syntax (like $var[2][3]), you have to put your expression inside {} braces.  In PHP 5, it is possible to actually make function calls within {} braces inside a HEREDOC string, but you have to go through a bit of work.  The function name itself has to be stored in a variable, and you have to call it like it is a dynamically-named function.  For example:  $fn = 'testfunction'; function testfunction() { return 'ok'; } $string = &lt;&lt;&lt; heredoc plain text and now a function: {$fn()} heredoc;   As you can see, this is a bit more messy than just:  $string = &lt;&lt;&lt; heredoc plain text and now a function: {testfunction()} heredoc;   There are other ways besides the first code example, such as breaking out of the HEREDOC to call the function, or reversing the issue and doing something like:  ?&gt; &lt;!-- directly output html and only breaking into php for the function --&gt; plain text and now a function: &lt;?PHP print testfunction(); ?&gt;   The latter has the disadvantage that the output is directly put into the output stream (unless I'm using output buffering), which might not be what I want.  So, the essence of my question is: is there a more elegant way to approach this?  Edit based on responses: It certainly does seem like some kind of template engine would make my life much easier, but it would require me basically invert my usual PHP style.  Not that that's a bad thing, but it explains my inertia..  I'm up for figuring out ways to make life easier though, so I'm looking into templates now. Why did Eve die with Maelon's data available? I saved Wrex in ME1, I saved Maelon's data in ME2, the import screen in ME3 confirmed this.    When I get to the missions on Tuchanka, I completed every one of the side missions before starting the mission where they take down the Reaper and distribute the genophage cure.  Then when I finally get to the lab, Mordin tells me that Eve has died.    Yet every site I've been to says that keeping the data allows her to live.  So why did she die?