Show that set of all invertible $n\times n$ matrices with real entries (denoted by $GL_n(R)$) is open set in $M_n(R)$.   My attempt: by open set I think it means neighborhood of every point in set is contained in set, but this definition doesn't seem to help solve this question. Any other approach?
I want to prove, that $GL(n,K)$ is open in $M(n,K)$, where ($K=\{\mathbb{R},\mathbb{C},\mathbb{H}\}$.  For the prove I don't want to use that the determinant is continous. Alternativly I assume a submultiplicative norm.  First step: Define $U_A:=\{B\in M(n,K):\Vert A-B\Vert&lt;\frac{1}{\Vert A^{-1}\Vert}\}$ and show that it is a neighbourhod of $A$ in $M(n,K)$.  Second step: I show that the series $\sum_{k\ge0}(1-A^{-1}B)^kA^{-1}$ converges absolutely and the limit is inverse to $B$.  For the second step I should use the Neumann series, right? But I have some problems with the first step. Any help?
What is the difference between the "jessie" and "jessie-updates" distributions in /etc/apt/sources.list? The  gives this example file:  deb http://httpredir.debian.org/debian jessie main deb-src http://httpredir.debian.org/debian jessie main  deb http://httpredir.debian.org/debian jessie-updates main deb-src http://httpredir.debian.org/debian jessie-updates main  deb http://security.debian.org/ jessie/updates main deb-src http://security.debian.org/ jessie/updates main   What is the difference between the jessie and jessie-updates distribution entries? Output URL of open firefox tabs in terminal How to make Latexmk use makeglossaries? Is there a way to connect my MacBook Pro to two external monitors (either VGA or HDMI; no Thunderbolt) in such a way that the Mac screen serves as a third monitor?  What hardware do I need? When I open the game, the screen stays at the Mojang sign and then crashes. It has something to do with my texture pack, but I can't start the game to change it and it won't let me delete the texture pack folder alone.  I've already deleted the bin and resources folders in the minecraft folder and downloaded the game again. I opened it but the same problem was still there.  How do I properly perform a clean uninstall of the game? display only 3 foreach result per row How much money did Charlie Harper have? This is quite tricky to figure out, and I am not looking for a specific number, but for a approximation.  As you know, Charlie (nearly) always had a lot of money to spend, but a few hints suggest that he actually did not have money at all (talking about absolute, not cash flow)   he went bancrupt after a few months without income he told Alan that there is no money, and everything just a house of cards he worries about Bertha giving away a few clothes he has three mortgages on his house when he died, there was no inheritage to Alan or anyone, and they had to sell the house ...   However, considering the lifestyle and spendings, Charlie must have had a substantial amount available.   driving the latest foreign expensive cars owning a malibu beachhouse paying the property tax for said house discussing retirement at 45 spending endless money on booze, betting and women ...   So all things considered, did he have money, or was he just a scam? siunitx uses math mode for symbols - workaround needed when using different fonts for math and text This is a follow up question from . The original problem has been solved using siunitx instead of dcolumn. However, the problem is that I am using mathpazo with osf as my main body font and eulervm as my math font. With siunitx used to decimal align tables this results in []() symbols being drawn in the math font, but the digits being drawn in the text font, as the following MWE example illustrates.  Is it possible to draw the symbols in the normal text font or redefine the mathfont for S columns completely?  \documentclass{scrartcl} \usepackage{booktabs} \usepackage[osf]{mathpazo} \usepackage[euler-digits]{eulervm}  % For Tables created by estout \newcommand{\sym}[1]{\ensuremath{^{#1}}} % for symbols in Table \usepackage{siunitx} \sisetup{ detect-mode,            group-digits            = false ,           input-signs             = ,           input-symbols           = ()[]-+ ,           input-open-uncertainty  = ,           input-close-uncertainty = ,           table-align-text-post   = false          } \begin{document} \begin{tabular*}{\textwidth}{     l @{\extracolsep\fill} *{2}{S[table-format=4.4]} @{}} \toprule \multicolumn{1}{@{}l}{\emph{Age}}  \\[0.5ex] 18--24 &amp;   [-1360.228]\sym{*}  &amp;  -1371.941\sym{*}  \\        &amp;       (6.915)         &amp;    (-7.953)        \\ 25--34 &amp;     -219.216          &amp;   -240.271         \\        &amp;     (514.894)         &amp;   (515.348)        \\ 35--44 &amp;     -310.715          &amp;   -311.438         \\        &amp;     (485.780)         &amp;   (486.013)        \\ 55$+$  &amp;       -5.135          &amp;     [2.656]        \\        &amp;      (53.267)         &amp;   (533.875)        \\ \bottomrule \end{tabular*} \end{document}   Edit: Thanks to David the problem has been solved including a fix for amsmath and extending to XeTeX. On a side note, the command for * generated by estoutshould now be simply plain text:  \newcommand{\sym}{#1} % for symbols in Table  \newcommand{\sym}[1]{\rlap{$#1$}} % Thanks to David Carlise again. How to create polygons with straight lines and right angles in QGIS? When creating a polygon, how do I do it so that the lines are straight (90, 0 degrees, etc.) and when I create corners they can be right angles? For instance, how do I create a perfect square or rectangle? Blacklight CFL Bulb Effect? I have been searching everywhere, and I can't find an answer so I'm coming here!  I'm trying to create a CFL Blacklight bulb. I've attached a couple images below of what I've tried to make. I'd love to have an effect like this:    Does anyone know how to recreate this? It seems like the color is affected by time as the bulb heats up, eventually being all white-ish. That would be fine, too, but I can't even seem to get the lighting effect of the pink glow around it and whatnot. I can post pictures of my node setup if you need it, though I think I made it necessarily complicated.  Thanks in advance!  UPDATE: This is what I have now:   I want to adjust the colors, but that isn't the issue now... The way I made this is a glass spiral that acts as the bulb. Inside of that bulb (that has a glass-like shader) is a purple material that I made that mimics the phosphor coating blacklight light bulbs have to filter light. Then, a separate mesh spiral is inside of that, and it has the emission stuff on it.   As Duarte suggested, I used the node setup he had for my emission, but used a transparent node instead of glossy. Why isn't the emission light from inside my glass causing the glass to be glossy? If I put a point light in the scene, it looks glossy.   Thanks in advance! I'm trying to find the name of a book I had as a kid back in the 80's/90's that was a collection of science fiction short stories and in particular one story in it.  The book: I remember the cover being mostly orange with the face of an alien taking up most of the cover. Scaly looking 'grey' style alien with hooded green eyes. They may have also been a 60's/70's style space ship in the background with a man and woman.  The story: Was set after all human life had ended in a great war and the story followed one of the many autonomous bombers still flying the skies carrying out their missions repeatedly bombing already leveled cities. At least that's how I remember it.  Any help/clues you can give would be much appreciated as I'd love to share these stories with my kid.  Some other stories it had, if any of these can be identified it might help me find the book:   Two guys talking in a room and one give the other a wooden carving of an apple and starts telling him the history of the object and how it's used to hypnotise people who hold it as they can't resist stroking it and it puts them in a catatonic state. A bunch of robots/machines wandering the post-apocalyptic country side, free of their duties to mankind. Their adventure comes to an end when they run into a hermit living in the country and their programming kicks in and they immediately become his servants. A story about first contact with an alien race, the aliens come to earth and land at the coordinates of an air-force base but upon arrival there is confusion as the humans cannot see the aliens and the aliens cannot see the humans. The aliens complain that they've been tricked and that their ship is sinking. The humans think this is strange until they realise the aliens are incredibly small and have landed in a puddle on the airfield. Let $T=\{z:|z|=1 \mathbin{\text{and}} z \in \mathbb C\}$ be the unit circle in the complex plane, considered as a topological group under complex multiplication and the usual topology. Show that every subgroup of $T$ is either dense in $T$ or finite.  How to prove the denseness? collapse vs minimize When I sum up the sizes of my files, I get one figure. If I run du, I get another figure. If I run du on all the files on my partition, it doesn't match what df claims is used. Why are there so many different figures for the total size of my files? Can't computers add?  Speaking of adding: when I add the “Used” and “Available” columns of df, I don't get the total figure. And that total figure is smaller than the size of my partition. And if I add up my partition sizes I don't get my disk size! What gives? Access a custom .NET DLL in VBScript I wrote a DLL in .NET and I want to access it in VBScript. I don't want to add it to the assembly directory.   Is there a way to point too the DLL and create an instance of it? New to the cloud so pardon this question as I'm certain it's relatively simple!  I've setup my Google Cloud Instance and I'm now trying to connect to it using FileZilla. I have a good understanding of how to do this... but I've been completely unable to download my .ppk file, public key, and private keys. The documentation clearly gives the folder that the files are located in.... but I'm unable to navigate to these folders (it's as if they don't exist) furthermore I'm not really sure how to download something to my computer through the SSH terminal.  Also somewhat infuriating is that I've found the keys in the web portal on Google Cloud however there is no option to download them. I can only view them in clear text.  Any help would be greatly appreciated! Rotating clears selected tags Changing the orientation of your device within the tag selection view while asking a question removes all selected tags and clears the textbox.     Steps to reproduce:     Ask a question (does not happen when editing) Edit tags Enter some text in the box  Select a tag from the suggestion list Repeat the previous two steps as many times as you care to Rotate the device (source / target orientation doesn't matter, just need to change it) All selected tags and text typed in the textbox are gone and must be re-entered   I've reproduced this using version 1.0.63 of the SE app on two devices:     HTC One E8 m2ss (the Hong Kong version) rooted and running stock 5.0.2 ATT Galaxy S3 i747, rooted and sim unlocked, running Slimkat 4.4.2.build.4.1.official ROM on Android 4.4.2 Getting my File Manager to link to my Google Drive I have a very small amount of storage on my computer and would like to conserve it, so I would like to know if there is a way to mount my Google Drive in Thunar or Nautilus on Ubuntu 12.04.5 LTS.   Yes, I know, it's weird that I'm saying that I have a small amount of storage when I have 2 file managers installed, but I literally have 5GB out of 16GB left. I am just learning ArcGIS 10.2 and it was working well until I tried to use the Export to a Geodatabase (single) option for a dbs table I made. Since then, it seems I now get the same Script Error for import/export functions. However, I just tried to export to Geodatabase (multiple) and error did not pop up. I spoke with tech support and they could not help me so far. I really do not know much about GIS so it is hard for me to fix this problem and cannot proceed with some things. Determine if $\sum_{n=1}^\infty \frac{n+4^n}{n+6^n}$ converges