How prove that the elementary operations don't change the rank of a matrix
Why do elementary matrix operations not affect the row space of a given matrix?
calculating expected number of packets. This is an exercise from textbook Analysis I by Amann/Escher.  Let $G$ be a group and $X$ a nonempty set. Then $G$ acts (from the left) on $X$ if there is a function $$G \times X \rightarrow X, \quad(g, x) \mapsto g \cdot x$$ such that: GA1: $e \cdot x=x$ for all $x \in X$. GA2: $g \cdot(h \cdot x)=(g h) \cdot x$ for all $g, h \in G$ and $x \in X$. Let $\mathrm{S}_{m}$ be the set of all permutations of $\{1,2,\cdots,m\}$. Then $(\mathrm{S}_{m},\circ)$ is a group where $\circ$ is function composition operation. Show that $$\mathrm{S}_{m} \times \mathbb{N}^{m} \rightarrow \mathbb{N}^{m}, \quad(\sigma, \alpha) \mapsto \sigma \cdot \alpha := \left(\alpha_{\sigma(1)}, \ldots, \alpha_{\sigma(m)}\right)$$ defines an action of $\mathrm{S}_{m}$ on $\mathbb{N}^{m}$.  I am stuck at proving the GA2, i.e. $\pi \cdot (\sigma \cdot \alpha) = (\pi \circ \sigma) \cdot \alpha$.  My attempt: Clearly, $\operatorname{id}$ is the identity element of $\mathrm{S}_{m}$ and $\sigma \cdot \alpha = \alpha \circ \sigma$. Because $\operatorname{id} \cdot \alpha = \left(\alpha_{\operatorname{id}(1)}, \ldots, \alpha_{\operatorname{id}(m)}\right) = \left(\alpha_1,\cdots,\alpha_m\right) = \alpha$, GA1 follows. For $\pi,\sigma \in \mathrm{S}_{m}$ and $\alpha \in \mathbb{N}^{m}$, we have $$\pi \cdot (\sigma \cdot \alpha) = (\alpha \circ \sigma) \circ \pi = \alpha \circ (\sigma \circ \pi) = (\sigma \circ \pi) \cdot \alpha$$  It is clear that $\sigma \circ \pi = \pi \circ \sigma$ if and only if $(\mathrm{S}_{m},\circ)$ is Abelian, which is not true in general. Please help me figure out where I am wrong! How do I move my website to a new server without interruption to email? Lattices are congruence-distributive When used like this:  import static com.showboy.Myclass;  public class Anotherclass{}   what's the difference between import static com.showboy.Myclass and import com.showboy.Myclass? Transparent background but solid title bar and text in Windows 7 I'd like to have a terminal like the one in Ubuntu where the background is transparent but title bar and text is still solid. I have tried Console, but setting the alpha value makes the entire window transparent, which makes the text hard to see.  Is that possible? Most Common Aspect Ratio? I'm building a web site that will allow users to upload photos. For design and layout purposes, I need to know the most common aspect ratios users will upload. What is the most common aspect ratio of photos (e.g from dSLR, smart phones, etc.)? Using the itemize environment inside a caption The following code does not compile. However, if I remove the itemize environment it does. I googled for this problem, and found suggestions that using the caption package would solve the problem, though I did not find any clear description of why the problem occurred in the first place. See for example    I'm using Debian squeeze with TeX Live 2009-11. The included version of caption has   \ProvidesPackage{caption}[2009/10/09 v3.1k Customizing captions (AR)]   I also tried downloading and using the caption package from     but this doesn't change anything. I put the sty files in my current directory. I assume that overrides the system files. This one corresponds to  \ProvidesPackage{caption}[2010/01/09 v3.1m Customizing captions (AR)]   Error message and example file follow.  *******************************************     ERROR: Argument of \@caption has an extra }.  --- TeX said --- &lt;inserted text&gt;                  \par  l.13     }  --- HELP --- From the .log file...  I've run across a `}' that doesn't seem to match anything. For example, `\def\a#1{...}' and `\a}' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `}' was spurious, just type `2' and it will go away. ******************************************************  \documentclass{article} \usepackage{caption} \begin{document}  \begin{figure}     \captionsetup{singlelinecheck=off}     \caption     {       foo       \begin{itemize}       \item bar       \end{itemize}     } \end{figure} \end{document} Search and replace a line in a file in Python I want to loop over the contents of a text file and do a search and replace on some lines and write the result back to the file. I could first load the whole file in memory and then write it back, but that probably is not the best way to do it.  What is the best way to do this, within the following code?  f = open(file) for line in f:     if line.contains('foo'):         newline = line.replace('foo', 'bar')         # how to write this newline back to the file Batch-Resizing images in OS X to a certain height Prove $(x+y)(y+z)(x+z)≤\frac8{27}(x+y+z)^3$ The entire exercise was like $$8xyz≤(x+y)(y+z)(x+z)≤\frac8{27}(x+y+z)^3\quad x,y,z&gt;0$$  and I managed to solve (demonstrate) the first inequality (from the left) using $$\frac{x+y}2≥\sqrt{xy}$$ but I can't solve the second one. I tried several ideas but I haven't succeeded. Thanks a lot. How do I make an HTML text box show a hint when empty? I want the search box on my web page to display the word "Search" in gray italics. When the box receives focus, it should look just like an empty text box. If there is already text in it, it should display the text normally (black, non-italics). This will help me avoid clutter by removing the label.  BTW, this is an on-page  search, so it has no button. I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ? How do I install ndiswrapper support for an unsupported RTL8723DE wireless module? I have not much experience with ndiswrapper but some vague memory of using it with some broadcom module long back. Now, here is the problem. Realtek RTL8723DE wireless module is used in my new HP 15-BS576TX laptop. I haven't installed Linux yet. But, would like to check if Ndiswrapper can help use the Windows driver to work in Linux. Here is the Windows 10/8/7 driver for this RTL8723DE wlan module:   I extracted the exe and can see a folder /RTWLANE_Driver/Wind7X86/ which has rtwlane.sys and few other files which I presume is related to the wireless driver. Is there any chances of luck with ndiswrapper considering the driver executable actually contains win8/win7 drivers.  Thank you. "My wife and I's seafood collaboration dinner" I just stumbled upon a :     My wife and I's seafood collaboration dinner. How does it look?   Sure enough, the top comment immediately points out that it should be "my wife's and my". However, a cross-post to the Grammar subreddit produced :     It's fine as it is written. "my wife and I" is a noun phrase, functioning as a subjective pronoun in the singular and made possessive with the apostrophe. It is exactly the same as "our".      It seems weird because you would never use "I's" on its own but it is not on its own here - it is part of a noun phrase.   That's a rather intriguing argument. Does it hold any water? Basically the question is how to get from this:   foo_id   foo_name 1        A 1        B 2        C   to this:   foo_id   foo_name 1        A B 2        C I want to render an image as a background image via an inline style in twig. I created a field called bg_image and attached it to the standard plain page.  After fiddling around for hours I was able to get the Image URL in node.html.twig  {{ file_url(node.field_bg_image.0.entity.uri.value) }}   but I could not get it to work inside field--field-bg_image.html.twig  Can I get the node from there anyhow, so I can then get the image?  How can I get the image URL to use as inline style? I thought maybe I can pass a variable from field--field-bg_image.html.twig to image.html.twig and then just render  {{ uri }}   instead of  &lt;img{{ attributes.addClass(classes) }} /&gt;   but I could not get it to pass the variable there unless I use includes  {% include 'image.html.twig' with {'image': image, 'isFromField': isFromField} %}   (isFromField is true, when it comes from the field--field-bg_image.html.twig) But that didn't work either. The Image was never rendered that way.  Would be very glad if you can help - my php knowledge is very basic. Thanks How would you explain JavaScript closures to someone with a knowledge of the concepts they consist of (for example functions, variables and the like), but does not understand closures themselves?  I have seen  given on Wikipedia, but unfortunately it did not help. tikz declare function and babel french option System.FinalException: Record is read-only: Trigger.updateCompetitors: line 24 I am writing a trigger to get all the Opportunity competitors into a field on Opportunity, however, getting an error while testing it.  Error: updateCompetitors: execution of AfterUpdate caused by: System.FinalException: Record is read-only: Trigger.updateCompetitors: line 24  Can you please check the below code and suggest me what's wrong with this?  trigger updateCompetitors on Opportunity(after insert, after update){    List&lt;Opportunity&gt; OppIds = new List&lt;Opportunity&gt;();     for (Opportunity opp: trigger.new){         OppIds.add(opp);     }  List&lt;OpportunityCompetitor&gt; oppComps = new List&lt;OpportunityCompetitor&gt;();  oppComps= [select id, CompetitorName, Strengths, Weaknesses from OpportunityCompetitor where OpportunityId in: Trigger.newMap.keySet()];  List&lt;Opportunity&gt; OppstoUpdate = new List&lt;Opportunity&gt;();   for (Opportunity opp: OppIds){     String allcomps = null;      if( oppComps.size()&gt;0 ) {       for (integer i = 0; i &lt; oppComps.size(); i++){          allcomps = allcomps + oppComps[i].CompetitorName+',';       }     }    opp.OpportunityCompetitors__c = allcomps;    OppstoUpdate.add(opp);  }    update OppstoUpdate; }