I am familiar with the case where the basis is not infinite (it becomes the case that it CANNOT be represented as such). However, many sources suggest that this is not the case for an infinite basis. How could one go about proving this?
I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field $\mathbf{F}$ is not a union of fewer than $|\mathbf{F}|$ proper subspaces" and possible generalizations.  A simple measure-theoretic proof over $\mathbb{R}$ is as follows: By countable additivity the sum of the measures of any collection of subspaces is zero since the measure of each subspace is zero, which is a contradiction.  I would like to look at proofs over arbitrary infinite fields and would like to know if similar statements hold for say modules (finitely generated or otherwise) over infinite rings.
Many LaTeX “hacks” begin with \makeatletter and end with \makeatother. What do these commands do? Cancelled UK visa on entry List files sorted by the number of lines they contain What does standard deviation tell us in non-normal distribution In a normal distribution, the  imparts standard deviation a lot of meaning, but what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions? Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ Possible Duplicate:          Proof to the formula  $$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$ I remember reading this as a kid and it starts out with:   a boy in his home village where the other kids are mean to him during sports and they heat up their home with floating logs or something.   The kid is being hunted by some reaper character and that reaper finds him at a place called "The Edge" or something similar.   He buys a glove covered in an expensive dust that purifies things and is really rare.  In one book they had to climb down the edge for some reason.  There was another part where people only communicated telepathically and when sneaking into somewhere he only thought of light.  One of the main things is the floating ships and how he is really good at flying them and he assembles a team for a mission or something.  There is like float powder and stuff.  I read this close to 10 years ago. Need Basic Viewer template for ArcGIS JSAPI 3.2/3.3 I am looking for basic viewer for ArcGIS JSAPI  but not  getting  proper sample/example.  I am going through  also but looks like its not working.  Going through  question also.  Checking out  example and getting confused that what is difference between Viewer template and layout?  So any help regarding this will be great! How can I check if share certificates I found have value? I have been clearing my parents flat and found share certificates from    P and O  European Ferries   Can I find out if they are still valid? If not, is there a market for them as collectibles or the like? Why is the PHP version different in phpinfo() and CLI? I realized that there is a miss matching between the PHP version reported by phpinfo() and php -v (in the CLI).  phpinfo():  5.5.24 php -v: 5.6.9   I am working on a Mac OS X 10.10 (Yosemite) and installed a library (php-version) to try to manage the PHP versions.  It reports me 3 different versions installed on my laptop:    5.4.41   5.5.25 * 5.6.9   I would like to set the same version of PHP in both Apache and CLI.  How can I tell Apache which PHP version use?  I tried to write this line in my httpd.conf:  LoadModule php5_module    /usr/local/opt/php56/libexec/apache2/libphp5.so   Then:  sudo apachectl restart   But all this not solve the problem… In my PHP info I still have: 5.5.24 How to set the background color of the Linux console screen? Note: I reworked the question to clarify the issues. This is a trivial question that can be answered in several parts because the question has several attributes.  Unfortunately, the question may be reworked when the person asking the question apprehends (knows) some parts of the answer as well as the structure of the question.  Therefore, the question is presented from a different point of view than the original one.  How to change the color setting¹ of the Linux console, using a shell utility? How to change the color setting¹ of a terminal emulator, using a shell utility?  Originally, I wanted to change the background color of the Linux console during the execution of a shell script. The following proposition works as I wanted.  #!/bin/sh  # set the background color of the terminal tput setab 6 clear  printf "%s\n" 'background: cyan'  # do some tasks  cd /usr/src/initramfs pwd ls -l  # the background of the shell output is cyan   Nonetheless, someone may encounter several glitches (or difficulties).   For instance, if the screen is not cleared, the background color is only applied to the text output, during a shell script execution. If the command tput setab 6 is submitted with the command line and the screen is not cleared, the background color is only applied to a small portion of the output text. If the command tput setab 6 is submitted on the command line and the screen is cleared, the background color is applied to the full screen but only persists until another command is submitted. [...]  ¹ background color I downloaded and installed Ubuntu from the official site. However, I don't know if I installed the 32-bit or 64-bit version.  In Windows 7 I could right click My Computer and it listed which version it was.  Is there an easy way to check in Ubuntu? Durrett Example 1.9 - Pairwise independence does not imply mutual independence? The example in question is from Rick Durrett's "Elementrary Probability for Applications", and the setup is something like this:     Let $A$ be the event "Alice and Betty have the same birthday", $B$ be the event "Betty and Carol have the same birthday", and $C$ be the event "Carol and Alice have the same birthday".   Durrett goes on to demonstrate that each pair is independent, since for example,   $P(A \cap B) = P(A)P(B)$.   However, he concludes that $A, B$, and $C$ are not independent, since   $P(A \cap B \cap C) = \frac{1}{365^2} \neq \frac{1}{365^3} = P(A)P(B)P(C)$.  I understand the reasoning here, and that one can generally show that arbitrary events $X$ and $Y$ are not independent by showing that $P(X\cap Y) \neq P(X)P(Y)$.   I am a little new to probability, though, and don't understand why exactly $P(A \cap B \cap C) = \frac{1}{365^2}$.   My progress so far:   I do see why $P(A) = P(B) =P(C) = \frac{1}{365}$, and thus why $P(A)P(B)P(C) = \frac{1}{365^3}$.  It seems like the sample space $\Omega = \{ (a, b, c) \mid a,b,c \in [365] \}$ -- i.e., all of the possible triples of numbers from 1 to 365, where 1 denotes January 1st, 2 denotes January 2nd, etc. From that, I can conclude $|\Omega| = 365^3$, but I'm not sure where to go from here.   It seems like once a single birthday is chosen, the rest are completely determined if they're all equal to each other - is this a good direction to go in? How to show textures in the 3D View editor? I have added some textures to objects in blender but the textures don't show up until I render the file. it can be difficult to know what the objects look like without rendering it. is there a way to have textures show in the editor even if they are at a lower res?  In editor:    Rendered: Dynamically creating keys in a JavaScript associative array All the documentation I've found so far is to update keys that are already created:  arr['key'] = val;  I have a string like this: &quot; name =  oscar &quot;  And I want to end up with something like this: { name: 'whatever' }  That is, split the string and get the first element, and then put that in a dictionary. Code var text = ' name = oscar ' var dict = new Array(); var keyValuePair = text.split(' = '); dict[ keyValuePair[0] ] = 'whatever'; alert( dict ); // Prints nothing. How do I set up a supply line? Overstaying on a US Visa Waiver Program A friend of mine came from Australia on the WVP and stayed 92 days. I made a mistake booking Her flight scheduling a Saturday arrival/departure at DFW. This amounted to a 92 day stay and with no funds to rebook a low cost flight through Qantas it was necessary to leave on the scheduled flight.   Question # 1) Was a automatic I-94W submitted by the airlines? She had no customs forms to fill out upon departure.   Question #2) How does this effect another WVP being issued if applied for? How to scale a tikzpicture to \textwidth Showing equality of sequence of holomorphic functions to limit function if converges uniformly locally I do not know how to proof following question:  Let $D$ be a domain in $C$.  ${{f}_{n}}$ be a sequence of holomorphic functions on $D$ that converges uniformly locally to a function non-constant $f$. Show that for each $z\in D$, there is a sequence ${{z}_{n}}\to z$ as ${{f}_{n}}({{z}_{n}})=f(z)$ for all sufficiently large $n$.  Actually, I know that I should use an assumption such as: ${{f}_{n}}({{z}_{n}})\ne f(z)$  Therefore, there is compact set    that  $\left\| {{f}_{{{n}_{k}}}}({{z}_{{{n}_{k}}}})-f(z) \right\|&gt;\varepsilon $  Then I think I have to use Montel theorem, but I do not know how to show the contradiction. We all know that in an idealised world all objects accelerate at the same rate when dropped regardless of their mass.  We also know that in reality (or more accurately, in air) a lead feather falls much faster than a duck's feather with exactly the same dimensions/structure etc.  A loose explanation is that the increased mass of the lead feather somehow defeats the air resistance more effectively than the duck's feather.  Is there a more formal mathematical explanation for why one falls faster than the other? Disable two column mode for separate part I am writing a document using the report class in two column mode, but I need a \section to spread across both columns, but how can I set that?  Something like  \documentclass[10pt,danish,a4paper,twoside,twocolumn]{report} \input{preamble.tex} \begin{document}  % This section should spread over both columns \section{Name}  \lipsum[1-3] \end{document}