If V and W are 3-dimensional subspace of $R^5$ then prove that V and W must have a nonzero common vector.
Prove that if $V$ and $W$ are three-dimensional subspaces of $\Bbb R^{5}$, then $V$ and $W$ must have a non-zero vector in common.  So far I got $V = \{v_{1}, v_{2}, v_{3}\}$ and $W = \{ w_{1}, w_{2}, w_{3}\}$ because they are 3-dimensional subspaces. Then $V \cup W = \{v_{1}, v_{2}, v_{3}, w_{1}, w_{2}, w_{3}\}$  My prof said that now I need to prove that they are linearly independent so this is a contradiction and the statement is wrong because we are in $\Bbb R^{5}$ and we have 6 elements on it.  I don't know how to express them as a linearly independent proof, so I hope anyone can give me a good explanation on it.
External hard drive and Recycle Bin behavior When my Seagate external hard drive is plugged in, its $RECYCLE.BIN folder automatically contains all the files contained in the Recycle Bin of the PC. Is it true that actually we don't have any Recycle Bin of external hard drive, rather it is the Recycle Bin of the PC? Why is it so? Where were the cannons in final fight in Wakanda in Infinity War? In Black Panther, Ulysses Klaue describes his arm cannon as a leaf blower in comparison to the cannons/weapons that Wakanda has.  Infinity War comes after the events of Black Panther, so the weapons should exist. During the final fight with the children of Thanos and Space Dogs (as Rocket refers to them), Wakandans could have used these powerful cannons, pointing them at the opened section of barrier. Instead they chose a fist fight.  Even M'Baku says that it would be the end of Wakanda, so the magnitude is huge and risks are very high. Why were these powerful weapons not used?    Edit 1: I watched the movie again and planes are there in the fight. But they were only used to destroy the spaceships that crashed. Which application does the vi command open? When you type vi in the terminal, the following output is shown  ~                             VIM - Vi IMproved                                 ~                                                                              ~                               version 7.3.547                                  ~                           by Bram Moolenaar et al.                             ~           Modified by pkg-vim-maintainers@lists.alioth.debian.org              ~                 Vim is open source and freely distributable                    ~                                                                                ~                        Help poor children in Uganda!                                                                                So according to this, vi must be launching VIm. But when you type  vim it gives  aditya@aditya-desktop:~$ vim The program 'vim' can be found in the following packages:  * vim  * vim-gnome  * vim-tiny  * vim-athena  * vim-gtk  * vim-nox Try: sudo apt-get install &lt;selected package&gt;   This shows that vim is not installed. Notably man vi and man vim launch the same man pages.  So what does the vi command actually launch? A theorem about Cesàro mean, related to Stolz-Cesàro theorem Original Title: Tauberian theorems and Cesàro sum  Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14)  $\newcommand\abs[1]{\left\lvert#1\right\rvert}$ If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\frac{s_0+s_1+\dotsb+s_n}{n+1}\qquad(n=0,1,2,\dotsc)$$ Put $a_n=s_n-s_{n-1}$ for $n\ge1$. Assume $M&lt;+\infty$ and $\abs{na_n}\le M$ for all $n$, and $\lim_{n\to\infty}\sigma_n=\sigma$, then $\lim_{n\to\infty}s_n=\sigma$.  The outline of the proof  If $m&lt;n$, then $$s_n-\sigma_n=\frac{m+1}{n-m}(\sigma_n-\sigma_m)+\frac1{n-m}\sum_{k=m+1}^n(s_n-s_k)\tag{*}$$ Notice that $\abs{s_n-s_k}\le(n-m-1)M\,/\,(m+2)$, fix $\epsilon&gt;0$ and associate with each $n$ the integer $m$ that satisfies $$m\le\frac{n-\epsilon}{1+\epsilon}&lt;m+1$$ Then $(m+1)\,/\,(n-m)\le1/\epsilon$ and $\abs{s_n-s_k}&lt;M\epsilon$. Hence $$\limsup_{n\to\infty}\,\abs{s_n-\sigma}\le M\epsilon$$  Questions and thoughts  It seems that the equation (*) comes out strangely. I wonder how to discover such kind of strange identities. So is there any observation, even deeper, to look through that equation?  Thanks! Can we add Markdown support for tables? How to prove that you have significant ties with family in your home country? How can I prove the immigrant officers that I have significant ties in my home country?  Are there any documents that I need? Pretty sure that my edit was correct but it was rejected? Abelian group (Commutative group) Prove that if in a group $(ab)^2= a^2 b^2$ then the group is commutative.  I am having a hard time doing this. Here is what I have so far: Proof:  $a^2 b^2= a^1 a^1 b^1 b^1$ =$aa^{-1}bb$ =ebb Hence,$aa^{-1}=e$  I am stuck, I do not know if this is the right process in proving this Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$ Ubuntu missing from HP boot menu I have a HP laptop with Windows 8.1 and Ubuntu 14.04 dual booted. I used to use F9 key to enter boot menu and chose Ubuntu and then GRUB would appear where I would chose Ubuntu again. This was working fine all this while. Either a hardware change or a Windows update has caused the Ubuntu option to go missing from the boot menu itself.  How do I fix this and boot into Ubuntu again? Can I view a list of my "great comments", perhaps in my profile? I'm a little surprised no one asked this yet, but why isn't it possible to show your comments in descending order of "great comment" votes it has received? You can do this with upvotes on questions and answers, and since I think comments are the third major activity on SO, I'd like to view them. Even if you couldn't order them, it should at least show the number of votes on the activity#comments page of your profile... Recent movie involving dimensions and time travel/displacement It's a (relatively) recent movie about a kid whose scientist father is missing or presumed dead.  The kid experiences strange situations (e.g., books from a case fall off).  In the end, it was the father the one that was throwing those books to draw the attention of the kid.  Does anyone know the name of this film? For $X$ and $Y$ random variables; $X$ follows the uniform distribution.  (1): if $Y=-\log X$  (2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}.  Why is this so? Intuitively statement (2) make sense to me. But i'd like a mathematical proof.  -Probably wrong working:  (1) seems to imply $\exp(-Y) = X$ (which is like saying $X$ is exponentially distributed, which is a contradiction, since its actually uniform!); or is it incorrect for me to do this since $X$ and $Y$ are random variables?  Ultimately how do I prove (2)?  Thanks In PHP you can just use $_POST for POST and $_GET for GET (Query string) variables. What's the equivalent in Python? I need to add a new field in the @book or @InProceedings entries: this field is "tome", for I have for instance the following title:  Francesco Bertolini, Società di trasmissione orale: mito e folclore, in Lo spazio letterario della Grecia antica, a cura di G. Cambiano, L. Canfora e D. Lanza, vol. I, tomo I, Salerno, Roma 1992, pp. 47--75.   So the following code is not enough:  @InProceedings{ bertolini:1992,   author                   = "Francesco Bertolini",   title                    = "Società di trasmissione orale: mito e folclore",   booktitle                = "Lo spazio letterario della Grecia antica",   editor                   = "G. Cambiano and L. Canfora and D. Lanza",   publisher                = "Salerno",   location                 = "Roma",   year                     = "1992",   volume                   = "1,   pages                    = "47--75", }   Using othes codes, I was able only to produce the "tome" field, but with a general trouble about the other fields:  \DeclareSourcemap{   \maps[datatype=bibtex]{     \map{       \step[fieldsource=tome,fieldtarget=usera]     }   } } \DeclareFieldFormat{usera}{tomo\space{#1}}   \DeclareBibliographyDriver{book}{%   \usebibmacro{author/editor}%   \setunit{\labelnamepunct}\newblock   \usebibmacro{title}%   \newunit\newblock   \usebibmacro{journal+issuetitle}%   \setunit{\bibpagespunct}%   \printfield{pages}%   \newunit\newblock   \printfield{usera}%   \usebibmacro{finentry}} \renewcommand*{\finentrypunct}{}   (useful for articles, but not for my case). By real, I'm not able to understand  which bibmascros I have to invocate...  I tried to write a biblatex-tome.dbx:  \ProvidesFile{biblatex-tome.dbx}   \DeclareDatamodelFields[type=field,datatype=literal,skipout=false]{tome}   \DeclareDatamodelEntryfields{tome}   Inside the main .tex file I put just this code:  \DeclareFieldFormat{tome}{\mkbibacro{tome}\addcolon\space #1} \newcommand*{\patchmoretome}[1]{   \xpatchbibdriver{#1}%     {\printfield{tome}}     {\printfield{tome}%           \newunit\newblock      \printfield{tome}}%     {\typeout{patching #1 to include more tome succeded}}%     {\typeout{patching #1 to include more tome failed}}% } % patch all drivers \patchmoretome{article} \patchmoretome{book} \patchmoretome{collection} \patchmoretome{inbook} \patchmoretome{incollection} \patchmoretome{inproceedings} \patchmoretome{manual} \patchmoretome{periodical} \patchmoretome{proceedings} \patchmoretome{report} \patchmoretome{thesis}   but it doesn't print the field in bibliographic entry. So I'm very confused about a (re)writing of driver Is it possible to install multiple instances of the same delphi service application? Invalid token 'while' in class, struct, or interface member declaration in very simple code I am not sure what the problem is but I keep receiving this error when I try to use a while statement in my code.     Invalid token 'while' in class,   struct, or interface member   declaration   I want to use a while loop to have something continuously update while a statement is true.  The rest of my code is rather long but whenever I type in the syntax:  while(a&lt;b) { //do whatever i want it to do here }   It gives me that compiler error right off the bat. Not quite sure what the problem is. I am doing this in a C# windows application under the Form1.cs file with all the other event handlers (for buttons and such).  Thanks!    I was unaware that loops had to be placed within a method (fairly new to c#), but I tried it and no errors were returned. Thanks for your help everybody!  Previously, I just had the loop within the main class of the program. Upgrade or replace a Wubi installation? I've got Windows 7 with Wubi 11.10 installed. I would like to upgrade to 12.04 but I'm not sure what the best path would be.  I'm not skilled enough to use Ubuntu all the time, and my iPhone requires me to run some Windows version for the damn iTunes software. I would love to run Windows XP but I can't figure out how to install it via USB -- my computer has no cd drive.  I'd like to run Windows XP and Wubi 12.04 on top of that. Or, perhaps, Ubuntu 12.04 natively and Windows XP alongside of that. So I guess I have 2 questions:   (lazy) Should I upgrade my Wubi from inside Ubuntu, or should I remove Wubi 11.10 from within Windows and install a brand new Wubi 12.04? (proper) How can I install Windows XP and Ubuntu 12.04 alongside each other?   Update: I am going to create a separate question post about the above #2 question... I realize it's wrong to ask two questions in one post when they are that different. I know that a similar question has been asked in:        although I didn't really understand the answer and couldn't really make it work.  However, I have tried something different. I decided to read the manual page for ls and it mentions different environment variables that can be set when dealing with the ls command. If one goes to the -G option it says to look at the CLICOLOR environment variables. I did and that one links you to documentation for the LSCOLORS environment variable. I went to it and it mentions:     "The value of this variable describes what color to use for which attribute when colors are enabled with CLICOLOR. This string is a concatenation of pairs of the format fb, where f is the foreground color and b is the background color. "   It also mentions that the default value of LSCOLORS is "exfxcxdxbxegedabagacad" and that each part specifies the color of some specific thing in the ls command. For example, the first pair ex in example specifies the color of directories. the second fx the color of symbolic links etc... e means blue and x means default "foreground".  1) First what does this mean: "when colors are enabled with CLICOLOR"?  2) What does "foreground" mean?  What must environment variables be for the ls command to color the output however I desire?    I also tried:  ls --color   But it threw me the following error:  ls: illegal option -- - usage: ls [-ABCFGHLOPRSTUWabcdefghiklmnopqrstuwx1] [file ...]   The same error happened when I did ls --color=auto: it alone also throws an error. I have no idea why, and am not sure if it was due to iTerm2 or OS X or why that happened. It seems it works on other systems... The ideal generated by $2$ and $ x$ in $\mathbb{Z}[x]$