Show that AB-BA = I is impossible for n x n matrices of real or complex numbers. Can someone help me with this problem?
The following question is from Artin's Algebra.      If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions.    I have no idea on how to tackle this question. I tried block multiplication, but it didn't appear to work.
Is FFmpeg missing from the official repositories in 14.04? I tried to install ffmpeg in trusty/Ubuntu 14.04 and got the following message:  $sudo apt-get install ffmpeg  Reading package lists... Done Building dependency tree        Reading state information... Done Package ffmpeg is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'ffmpeg' has no installation candidate   Why isn't the package available? How to change the default views of quad view? How do I change the TOP ortho view to BACK ortho view for my BACK image?   check out the pic I uploaded to see what I'm talking about. What is the difference in meaning among these sentences? Walking is good exercise. Walking is a good exercise. Walking is a good form of exercise.     I'd like to know the difference in meaning among these sentences. Thanks a lot. Implementing toString on Java enums I am writing a bash script to look for a file if it doesn't exist then create it and append this to it:  Host localhost     ForwardAgent yes   So "line then new line 'tab' then text" I think its a sensitive format.  I know you can do this:  cat temp.txt &gt;&gt; data.txt   But it seems weird since its two lines. Is there a way to append that in this format:  echo "hello" &gt;&gt; greetings.txt Changing email addresses on my @me.com and @icloud.com What does size of a directory mean in output of 'ls -l' command? I am reading the Dummit and Foote's proof of Cauchy's theorem for abelian groups. On the highlighted line, how do you conclude from $y\notin N$ and $y^p\in N$ that $\langle y^p \rangle \neq \langle y \rangle$ and that $|y^p|&lt;|y|$?     Proposition 21.   If $G$ is a finite abelian group and $p$ is a prime dividing $|G|$, then $G$ contains an element of order $p$.      Proof:   The proof proceeds by induction on $|G|$, namely, we assume the result is valid for every group whose order is strictly smaller than the order of $G$ and then prove the result valid for $G$ (this is sometimes referred to as complete induction).   Since $|G| &gt; 1$, there is an element $x \in G$ with $x \neq 1$.   If $|G| = p$ then $x$ has order $p$ by Lagrange’s Theorem and we are done.   We may therefore assume $|G| &gt; p$.      Suppose $p$ divides $|x|$ and write $|x| = pn$.   By Proposition 2.5(3), $|x^n| = p$, and again we have an element of order $p$.   We may therefore assume $p$ does not divides $|x|$.      Let $N = \langle x \rangle$.   Since $G$ is abelian, $N \trianglelefteq G$.   By Lagrange’s Theorem, $|G/N| = |G|/|N|$ and since $N \neq 1$, $|G/N| &lt; |G|$.   Since $p$ does not divide $|N|$, we must have $p \mid |G/N|$.   We can now apply the induction assumption to the smaller group $G/N$ to conclude it contains an element $\bar{y} = yN$, of order $p$.   Since $y \notin N$ ($\bar{y} \neq \bar{1}$) but $y^p \in N$ ($\bar{y}^p = \bar{1}$) we must have $\langle y^p \rangle \neq \langle y \rangle$, that is, $|y^p| &lt; |y|$.   Proposition 2.5(2) implies $p \mid |y|$.   We are now in the situation described in the preceding paragraph, so that argument again produces an element of order $p$.   The induction is complete.      (Original image .) addplot3[surf] creates lines instead of a surface I have been trying to create a surface plot but I am unable to get it to work. Any help would be greatly appreciated. I've tried using surf which only produces lines instead of a surface (Fig. 1). Note: Although I was originally able to produce output, thesurf plot no longer compiles: "Package tikz Error". I've tried compiling with both pdflatex and lualatex with -shell-escape flag.    Figure 1  I also tried to use the patch style plot. This produces gibberish (Fig. 2)    Figure 2  MWE:  \documentclass{article} \usepackage{pgfplots} \usepgfplotslibrary{external} \usepgfplotslibrary{patchplots} \tikzexternalize  \begin{document} \begin{figure}[t]   \centering    \tikzsetnextfilename{surface_plot}   \begin{tikzpicture}     \begin{axis} [       height=12cm,       width=15cm,       grid=major,       grid style={dotted},       colormap/jet,       mesh/ordering=x varies     ]       %\addplot3[patch,patch type=biquadratic,shader=faceted interp,patch refines=3] coordinates {       \addplot3[surf] coordinates {  (-6,-6,446.788)  (-5,-6,423.361)  (-4,-6,413.264)  (-3,-6,402.349)  (-2,-6,388.948)  (-1,-6,391.747)  (0,-6,386.874)  (1,-6,423.778)  (2,-6, 439.19)  (3,-6,449.074)  (4,-6,462.155)  (5,-6,484.399)  (6,-6,505.412) (-6,-5, 423.48)  (-5,-5,392.382)  (-4,-5,406.696)  (-3,-5,392.592)  (-2,-5,371.158)  (-1,-5,384.142)  (0,-5,394.666)  (1,-5,408.105)  (2,-5, 422.51)  (3,-5,420.173)  (4,-5,445.816)  (5,-5,464.731)  (6,-5,485.349) (-6,-4,402.646)  (-5,-4,371.879)  (-4,-4,376.239)  (-3,-4,396.874)  (-2,-4,382.761)  (-1,-4,386.665)  (0,-4,397.148)  (1,-4,410.875)  (2,-4,425.998)  (3,-4,385.143)  (4,-4,420.458)  (5,-4,438.432)  (6,-4,    458) (-6,-3,423.219)  (-5,-3,370.582)  (-4,-3,464.786)  (-3,-3,443.921)  (-2,-3,425.196)  (-1,-3,427.138)  (0,-3,438.002)  (1,-3, 453.97)  (2,-3,473.021)  (3,-3,418.583)  (4,-3,435.887)  (5,-3,414.993)  (6,-3,475.148) (-6,-2,401.995)  (-5,-2,385.612)  (-4,-2,438.956)  (-3,-2,418.583)  (-2,-2,400.093)  (-1,-2,400.773)  (0,-2, 410.58)  (1,-2,426.024)  (2,-2, 444.92)  (3,-2,397.131)  (4,-2,413.739)  (5,-2,431.741)  (6,-2,451.267) (-6,-1,382.978)  (-5,-1,335.274)  (-4,-1,416.598)  (-3,-1,396.718)  (-2,-1,378.348)  (-1,-1,377.455)  (0,-1,32.0089)  (1,-1,400.683)  (2,-1,419.918)  (3,-1,377.371)  (4,-1,393.286)  (5,-1,374.648)  (6,-1,429.077) (-6, 0,365.902)  (-5, 0,350.965)  (-4, 0,397.527)  (-3, 0, 378.36)  (-2, 0,360.454)  (-1, 0,358.447)  (0, 0,    363)  (1, 0,381.147)  (2, 0,401.177)  (3, 0,362.155)  (4, 0,378.174)  (5, 0,395.513)  (6, 0, 414.34) (-6, 1,372.511)  (-5, 1,326.945)  (-4, 1,407.215)  (-3, 1,388.646)  (-2, 1,371.945)  (-1, 1,371.003)  (0, 1,380.027)  (1, 1,396.414)  (2, 1,416.602)  (3, 1,374.897)  (4, 1,391.276)  (5, 1, 373.32)  (6, 1,428.284) (-6, 2,391.149)  (-5, 2,376.191)  (-4, 2,406.941)  (-3, 2,369.633)  (-2, 2,393.583)  (-1, 2,394.184)  (0, 2,404.263)  (1, 2,420.588)  (2, 2,  440.5)  (3, 2,416.631)  (4, 2,374.948)  (5, 2,391.855)  (6, 2,418.511) (-6, 3,375.967)  (-5, 3,361.698)  (-4, 3,357.421)  (-3, 3,368.273)  (-2, 3,377.106)  (-1, 3,378.664)  (0, 3,369.175)  (1, 3, 382.42)  (2, 3,397.821)  (3, 3,377.942)  (4, 3,384.944)  (5, 3,387.862)  (6, 3,424.429) (-6, 4,396.478)  (-5, 4,381.554)  (-4, 4,376.162)  (-3, 4,388.843)  (-2, 4,375.881)  (-1, 4,378.384)  (0, 4, 355.14)  (1, 4,403.654)  (2, 4,419.779)  (3, 4,398.599)  (4, 4,407.113)  (5, 4,426.715)  (6, 4,448.245) (-6, 5,409.364)  (-5, 5,380.492)  (-4, 5,387.843)  (-3, 5,360.034)  (-2, 5, 362.23)  (-1, 5,364.909)  (0, 5,358.321)  (1, 5,363.226)  (2, 5,379.325)  (3, 5,397.152)  (4, 5,410.086)  (5, 5,433.909)  (6, 5,460.548) (-6, 6, 423.99)  (-5, 6,408.967)  (-4, 6,400.475)  (-3, 6,372.922)  (-2, 6,371.116)  (-1, 6, 373.92)  (0, 6,384.778)  (1, 6,  398.8)  (2, 6, 415.62)  (3, 6,420.127)  (4, 6,436.064)  (5, 6,461.957)  (6, 6,491.123)        };     \end{axis}   \end{tikzpicture}      \end{figure} \end{document}   Thank you so much. Showing that $Q_n=D_n+D_{n-1}$ Let $T_n$ be the set of permutations of $\{1,2,\ldots,n\}$ which do not have $i$ immediately followed by $i+1$ for $1\le i\le n-1$; in other words, let \begin{align} T_n=\{\sigma \in S_n: \sigma(i)+1\ne\sigma(i+1) \text{ for all } 1\le i\le n-1\} . \end{align} Let $Q_n$ be the number of elements of $T_n$.    Let $D_n$ be the number of  of $\{1,2,\ldots,n\}$.  It is not hard to show algebraically that $Q_n=D_n+D_{n-1}$, but I am having difficulty coming up with a combinatorial argument to show why this is true.  (I believe there are $D_{n-1}$ elements of $T_n$ that leave $n$ fixed, and $D_n$ elements of $T_n$ that move $n$, but I don't know how to justify this combinatorially.) "To do this" or "For doing this" to start explaining something? Many questions have been asked and answered about "to + bare infinitive" vs "for + gerund" in the middle of a sentence.  However, when one wants to start explaining something, what is better? For instance:  "Now, let us focus on baking the bread.   To do this, start by turning the oven on.", or For doing this, start by turning the oven on."?   Are there even better expressions? I have not found anything concerning this situation, and I feel that the answers about the same problem but in the middle of a sentence do not apply here. Use abbreviations for 1000+ score in linked posts I don't really enjoy the ragged view of linked posts:    This feature request is to use a format that suggested in ; something that matches the user-profile tags page in terms of the tally:    That is, use abbreviations for 1000+ (k for 1,000+, m for 1,000,000+, ...). Simplify a combinatorial sum Why is $\sum_{k=0}^n (_k^n) 2^k$ simplified as $3^n$ and not just $2^n$ ? The answer uses the binomial theorem as a solution: let $a=1$ and let $b=2$. So $1+2 = 3$. But since $1+2 = 3$ why do we not use $1+1 = 2$? What is the word for the emotion I feel when I see someone being humiliated? When I see someone else being embarrassed / humiliated, for example a singer forgetting their words in a concert, I would say something like "I feel embarrassed for them". But is there a better word to describe this feeling?  I would say it's a mix of:   Empathetic discomfort for someone else's humiliation Tension; hope that spectators will be forgiving Frustration that I cannot help them   Other scenarios I would feel the same emotion:   Seeing someone make a presentation, the slides are all out of order, and their face gets red with embarrassment while there's a tense silence in the room. Seeing someone make an embarrassing social blunder in front of unforgiving people. Seeing a speaker who keeps making a distracting and annoying gesture, and they don't realize that the whole audience is annoyed. Seeing someone be taken advantage of, like seeing someone naively and happily accept a terrible offer on a car. Seeing a person of strong image (for example the CEO of your company) do something like accidentally leave their webcam on after a conference call, and employees see him/her let out a big burp and pick their nose or something.   In these cases it's important that the audience is not forgiving, even if they don't confront the person about it. The person may or may not know they are being humiliated.  "Pity" as proposed in an answer seems very close, but I think it lacks a few components. I think "pity" has a chronic connotation - that you are referring to the general state of things rather than a specific event.  I think "pity" also depends on the subject suffering, but in my example they are often oblivious.  It could very well be that there is no word for this at all.  NOTE: Some people do not feel the same way I do in this scenario, so you may not think this question makes any sense. Japan is a volcanically active country which means it has a plethora of onsen (hot springs) all around the country.  Visiting these onsen is supposed to be a popular vacation destination for the work-weary Japanese.    I've been told that, as with other things in Japan, there is a rather strict etiquette surrounding the use of the onsen.  In particular, the two things I've read about so far are   Swimming Suits are not allowed. It is very important that one bathe immediately before entering the onsen.   Are these true?  Are there any other taboos that a naive American should be aware of? latex table border dont appear / to big / to small still working on that table. Here's my new question reguarding the table borders:  It appears, that i do something wrong setting boarders. Here is my code, and a screenshot:  \documentclass{article} \usepackage{multirow} % Used for the vertical cells \usepackage{rotating} % Used for the vertical cells  \usepackage{xcolor} % Used for cell-colors \usepackage{colortbl} % Used for cell-colors  \usepackage[a4paper]{geometry}% Used to set the pdf margins manually \newgeometry{left=2cm,bottom=0.1cm}  \definecolor{grey}{HTML}{F2F2F2} % Define grey for the cells \definecolor{red}{HTML}{FF0000} % Define red for testing  \begin{document} \scriptsize % Use a small font size  \begin{tabular}{|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0,6cm}|}  \cline{10-17}  \multicolumn{9}{|c|}{} &amp; \multicolumn{2}{p{\dimexpr (0.6cm)*2+2\arrayrulewidth+2\tabcolsep\relax}|}{\cellcolor{grey}Work to retired ratio} &amp; \multicolumn{4}{p{\dimexpr (0.6cm)*4+1\arrayrulewidth+2\tabcolsep\relax}|}{Health care sizing (per thousand)} &amp; \multicolumn{2}{p{\dimexpr (0.6cm)*2+1\arrayrulewidth+2\tabcolsep\relax}|}{\cellcolor{grey}Inverse dependency ratio}\\\cline{3-9}  \multicolumn{2}{|c|}{} &amp; \multicolumn{4}{c|}{\cellcolor{grey}Population (mio.)} &amp; \multicolumn{3}{c}{Workforce (1000)}  &amp; \multicolumn{2}{|c|}{\cellcolor{grey}} &amp; \multicolumn{2}{l|}{Physicians by} &amp; \multicolumn{2}{l}{Nurses by} &amp; \multicolumn{2}{|l|}{\cellcolor{grey}Old agers by}\\\hline  \multicolumn{2}{|c|}{} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Overall}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Young age}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\cellcolor{grey}\parbox{4cm}{Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Overall}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Physicians (Proj: 2025)}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Nurses (Proj: 2030)}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}Working age by Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}Work force by Old age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Workforce}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Workforce}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{Working age}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}1 physicians}\end{sideways} &amp; \begin{sideways}\parbox{4cm}{\cellcolor{grey}1 nurse}\end{sideways}\\\hline  \multicolumn{2}{|l}{USA} &amp; \multicolumn{4}{l}{\cellcolor{grey}} &amp; \multicolumn{3}{l}{} &amp; \multicolumn{2}{l}{\cellcolor{grey}} &amp; \multicolumn{4}{l}{} &amp; \multicolumn{2}{l|}{\cellcolor{grey}}\\\hline  1990/1 &amp; &amp; 248,6 &amp; 71,3 &amp; 146,1 &amp; 31,2 &amp; 127.500 &amp; 615,4 &amp; 1491,6 &amp; 4,68 &amp; 4,09 &amp; 4,83 &amp; 4,21 &amp; 11,70 &amp; 10,21 &amp; 50,7 &amp; 20,9\\\hline  1 &amp; 2 &amp; 3 \\\hline  \end{tabular}  \end{document}     On 1 the Top border of Population is not visible although i did used \\\cline{3-9}\cline{12-17} on that line.  In 2 both borders are pushed a bit to the right for no reason i can see.  And in 3 again the top borders are not visible despite using \cline{10-17} on that line. What did i do wrong?  Thanks! Is there a way to see questions with a certain tag from all Stack Exchange sites? For example, I see there are questions with the  tag on Stack Overflow, but there are also such questions on Super User, Unix/Linux, and other sites in the network.  Is there a way to follow a tag across all of the Stack Exchange sites, so that one can see the global activity of a topic without visiting all the sites separately? Putting char into a java string for each N characters How can I make calls related to a case display in its Chatter feed? I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? How to pull email address and password from text file with CMD? I have a text file named file.txt Which contains,  Combo: testing123mail@gmail.com:password As Combo:  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf some text here some more text here As Combo: hadvsfhia@ahfjbadj.com:password@1 some random text here iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   I am running the below code, to filter only mail:pass from the above text file file.txt  for /f "tokens=3" %%a in ('type file.txt^|find "As Combo:"') do &gt;&gt;mail.txt echo %%a for /f "tokens=2" %%b in ('type file.txt^|find "Combo:"') do &gt;&gt;mail.txt echo %%b for /f "tokens=1" %%c in ('type file.txt^|find ":"') do &gt;&gt;mail.txt echo %%c   Expected Output in mail.txt,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   But, am getting actual output is,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password Combo: Combo: Combo: As As iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   Note: The mail id and passwords will be differ in different text file. Above given mail id and passwords are typed by me randomly for example.  Kindly fix my code and help to get expected output. Thanks in advance.