I want to calculate the size of $GL_n(\mathbb{F}_p)$. Originally I was locking specifically for the size of $GL_2(\mathbb{F}_5)$ but I managed to calculate it by directly calculating ${\{A\in M_2(\mathbb{F}_5)\mid detA=0\}}$ and some arithmetic. I was wondering if there is a generalization and couldn't find one. Thanks in advance!
$F$ is a finite field of order $q$. What is the size of $\operatorname{GL}_n(F)$ ?  I am reading Dummit and Foote "Abstract Algebra".  The following formula is given:  $(q^n - 1)(q^n - q)\cdots(q^n - q^{n-1})$.  The case for $n = 1$ is trivial.  I understand that for $n = 2$  the first row of the matrix can be any ordered pair of field elements except for $0,0$. and the second row can be any ordered pair of field elements that is not a multiple of the first row.  So for $n = 2$  there are $(q^n - 1)(q^n - q)$ invertible matrices.  For $n\geq 3$, I cannot seem to understand why  the formula works.  I have looked at Sloane's OEIS A002884.  I have also constructed and stared at a list of all $168$ $3\times 3$ invertible matrices over $GF(2)$.  I would most appreciate a concrete and detailed explanation of how say $(2^3 - 1)(2^3 - 2)(2^3 - 2^2)$ counts these $168$ matrices.
How do I "parent" objects? How do I tackle this combinatorics problem about married couples around a table? How do I push amended commit to the remote Git repository? When I've worked a bit with my source code, I did my usual thing commit and then I pushed to a remote repository. But then I noticed I forgot to organize my imports in the source code. So I do the amend command to replace the previous commit:  &gt; git commit --amend   Unfortunately the commit can't be pushed back to the repository. It is rejected like this:  &gt; git push origin To //my.remote.repo.com/stuff.git/  ! [rejected]        master -&gt; master (non-fast forward) error: failed to push some refs to '//my.remote.repo.com/stuff.git/'   What should I do? (I can access the remote repository.) Center figure that is wider than \textwidth I have a figure that is wider than the \textwidth of my document. (I don't want to change the \textwidth.) As a result, the leftmost part of the figure is flush with the left margin, while the rightmost part spills into the right margin too much. I'd like to center the figure, so that it will exceed the \textwidth equally on both sides. Why is the CDF of a sample uniformly distributed I read  that given a sample $ X_1,X_2,...,X_n $ from a continuous distribution with cdf $ F_X $, the sample corresponding to $ U_i = F_X(X_i) $ follows a standard uniform distribution.  I have verified this using qualitative simulations in Python, and I was easily able to verify the relationship.  import matplotlib.pyplot as plt import scipy.stats  xs = scipy.stats.norm.rvs(5, 2, 10000)  fig, axes = plt.subplots(1, 2, figsize=(9, 3)) axes[0].hist(xs, bins=50) axes[0].set_title("Samples") axes[1].hist(     scipy.stats.norm.cdf(xs, 5, 2),     bins=50 ) axes[1].set_title("CDF(samples)")   Resulting in the following plot:    I am unable to grasp why this happens. I assume it has to do with the definition of the CDF and it's relationship to the PDF, but I am missing something...  I would appreciate it if someone could point me to some reading on the subject or help me get some intuition on the subject.  EDIT: The CDF looks like this: Would someone give a thorough explanation of 1.14 Minecraft's /data modify? Okay, so I've been searching all over the internet for an explanation of data modify ever since I heard Sethbling passively mention it in one of his videos. The only  thing I could find on it was the wiki and the update log for the 1.14 snapshot. So, I checked these, and they hardly explained it at all. I can get /data modify block 123 20 123... and that's as far as I get.  Minecraft wiki does not at all define clearly what  is (the thing meant to go after coords), nor does it specify what each operation does to what. Operations include (without explanation, and excluding set (which it does explain)): append, insert, merge, and prepend. Then, it has from|value as the next option, which aren't explained either. After this, it just has block/entity and coords/id. So, I know the format (mostly), but I haven't the slightest idea what any of it means. And nothing I've seen on the internet even mentions the command save for Minecraft wiki itself.  Please explain if you know what data modify is. Thanks! I often find myself wanting to install something, but I'm unsure of what package it's in. This is a very common occurrence for me:  $ make html sphinx-build -b djangohtml -d _build/doctrees   . _build/html make: sphinx-build: Command not found make: *** [html] Error 127  $ sudo apt-get install sphinx E: Unable to locate package sphinx  $ sudo apt-get install sphinx-build E: Unable to locate package sphinx-build   googles  $ sudo apt-get install python-sphinx   Is there a better way? Thousands of .log files in Exchange 2007 Server folder I'm a bit of an Exchange admin noob so I don't know if this is normal, but on our Exchange 2007 server under C:\Program Files\Microsoft\Exchange Server\Mailbox\First Storage Group\ there are 30,000 1MB .log files.  I don't know if those should be there or not, but they keep multiplying like little jackrabbits.  I've so far resisted deleting them as I'm sure something would break, but it's come to the point now where our 80GB disk is completely full up and Exchange has stopped receiving messages.  My best guess is that these are transaction logs and would be useful in rebuilding the database should something go awry, but 30GB of transaction logging seems a bit excessive.  Is there a way to limit it or is that even a good idea?  And of course did I completely miss the point? I am trying to slice up my models using the Boolean modifier.  After applying the modifier on my smooth mesh, I get the expected result, a flat dissection to separate my mesh into two.  The problem lies in how I can resolve the visual "jaggered" edge that separates my (smooth) mesh exterior from the flat dissection face.    Is it possible to achieve a smooth-to-flat variation without this artifact?   I tried to alter a mixture of the two but end up with artifacts, or unwanted blocky flat mesh faces around the dissection edge of my mesh. I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:     DEFINITION Given  $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x &lt; z+1$$ and we denote it by $[x]$.   Now he asks to prove some basic things about it, such as: if $n\in \Bbb Z$, then $[x+n]=[x]+n$  So I proved it like this: Let $z=[x+n]$ and $z'=[x]$. Then we have that  $$z\leq x+n&lt;z+1$$  $$z'\leq x&lt;z'+1$$  Then $$z'+n\leq x+n&lt;z'+n+1$$  But since $z'$ is an integer, so is $z'+n$. Since $z$ is unique, it must be that $z'+n=z$.  However, this doesn't seem to get me anywhere to prove that  $$\left[ {2x} \right] = \left[ x \right] + \left[ {x + \frac{1}{2}} \right]$$  in and in general that   $$\left[ {nx} \right] = \sum\limits_{k = 0}^{n - 1} {\left[ {x + \frac{k}{n}} \right]} $$  Obviously one could do an informal proof thinking about "the carries", but that's not the idea, let alone how tedious it would be. Maybe there is some easier or clearer characterization of $[x]$ in terms of $x$ to work this out.  Another property is $$[-x]=\begin{cases}-[x]\text{ ; if }x\in \Bbb Z \cr-[x]-1 \text{ ; otherwise}\end{cases}$$  I argue: if $x\in\Bbb Z$, it is clear $[x]=x$. Then $-[x]=-x$, and $-[x]\in \Bbb Z$ so $[-[x]]=-[x]=[-x]$. For the other, I guess one could say:  $$n \leqslant x &lt; n + 1 \Rightarrow  - n - 1 &lt; x \leqslant -n$$  and since $x$ is not an integer, this should be the same as $$ - n - 1 \leqslant -x &lt; -n$$  $$ - n - 1 \leqslant -x &lt; (-n-1)+1$$  So $[-x]=-[x]-1$ I have an argument with the conclusion: 'The sky is blue and it is not the case that the sky is blue'.  Without knowing the premises of this argument, am I right in saying this argument is invalid since the this conclusion is necessarily false? What is the purpose of the Assemblies node in Web.Config? When creating a new ASP.NET AJAX Web Application in Visual Studio 2005, the default web.config contains the following section (inside the compilation node):  &lt;assemblies&gt;     &lt;add assembly="System.Web.Extensions, Version=1.0.61025.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35"/&gt; &lt;/assemblies&gt;   Whether or not the assemblies node exists doesn't seem to affect the AJAX functionality.  Update panels, etc work fine regardless of whether or not the reference to System.Web.Extensions is included in Web.Config.  What's the purpose of this section, and can it be safely removed? Safari autofill multiple e-mail adresses Is there a way to configure your Stack Overflow account so that you receive notification emails when posts are made to specific tags?    Or do you simply have to log on to see new posts? Where does top command get all the data for CPU Utilization? The Android app crashes when trying to upload a picture from my phone Which focal-length lens is usually used for portrait photography, and why? What focal length or range of focal lengths are usually used for portraits?   What is used by professionals? Why would one select a certain focal length? What are the advantages and disadvantages of different choices, and are there certain situations where an unusual choice might be better? Just received ~70 unsolicited password reset mails from Super User, so abusive... Remove a child with a specific attribute, in SimpleXML for PHP Why does holding something up cost energy while no work is being done? I read the definition of  as  $$W ~=~ \vec{F} \cdot \vec{d}$$  $$\text{ Work  =  (Force) $\cdot$ (Distance)}.$$ If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going?