Let U and w be two subspaces of the same vector space V.Is U union W necessarily a vector space?
I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
How can I make a single event handler handle ALL Button.Click events? Ready to use leaderboard service. Is any? I'm looking for ready to use thirdparty leaderboard service where player's achivementes and scores can be placed. I'd intagrate it into my game to allow players push scores and locate themselves in the tables of tops. It would be nice whether anybody experienced share some points with me. Change keyboard layout (English UK) on command line to English US Object rotate in a strange way today i run into another problem. i dont know how i have done it (probably i clicked sth what i should not click) - anyway - since that moment my objects are behaving totally different when i try to rotate them. the objects are now rotating around the pivot point but they keep their alignment - not sure if i am clear. eg. if u would rotate a plane which is lies flat on the ground it makes the rotation but the faces of the plane would still be up and dont which shouldn't be the case with the "normal" rotation that i know. so i want to understand what i have done (because this kind of rotation could be also usefull sometimes for me) and how can i return it in the case it happens again. Installed Ubuntu 13.04 on a Dell laptop I had laying around.  Install goes well but when I'm done, I only get a command line, no GUI at all.  Tried several versions, tried 32 vs 64 bit all give me the same result.  Any ideas on what to do/try? Inserting a term into a custom taxonomy Idempotent linear operators are projections I'm working on this problem: Suppose $V$ is a Hilbert space and $P : V \to V$ is a linear map such that $P^2= P$ and $\Vert Pf \Vert \leq \Vert f \Vert$ for every $f \in V$. Prove that there exists a closed subspace $U$ of $V$ such that $P = P_U$. I did do some searching online and I know that $P$ is an idempotent matrix and I know that $U$ is supposed to be $\text{range}P$, which I can show is closed. However, I'm having a hard time showing $P_{\text{range}P} = P$. I know that I can decompose $f = g + h$, where $g \in \text{range}P$ and $h$ is in the orthogonal complement of $\text{range}P$. I'm not sure how to proceed. Most of the online searches require that $P$ be self-adjoint, but this question doesn't have this condition. Can I get some help? Defining macros with arguments With newcommand I can define customized command like \ABC. Is there any way to define a command with arguments. For example I want \brat{A}{B} be the same as [\![A,B]\!]. Although we can create or update fieldsets using salesforce UI but Is there any way to insert or update FieldSets using apex code (using metadata or API calls) ? LM317 Heat sink, how big? UK PLAB Visa Refusal due to unexplained bank deposits I had applied for a PLAB (Doctors) Visa for the UK to appear in PLAB test. I got refused on the basis of two recent large deposits in the bank statement I provided (my mothers). In reality, these are my own savings which I had deposited in her account to make it more strong. The only objection I got was that I must provide evidence for the source of these deposits. Other than this, I have previously travelled to the UK and USA as well as other parts of the world and my travel history is fine. Can someone please advise on how to proceed and re-apply? How can the following inequation be proven?  $$a^2 + b^2 + c^2 \ge ab + bc + ca$$ How do I set read/write permissions my hard drives? How to Hide sections of InfoPath form based on Group membership I need to Hide "Sections" of InfoPath form using SharePoint or AD group membership (either is fine). How can I achieve this?  InfoPath 2010 in SharePoint 2010 make tail -f exit on a broken pipe I want to watch a file until a bit of text appears  I found this answer:   but when I tried it on ubuntu, it doesn't exit:  $ echo one &gt; test.txt $ echo two &gt;&gt; test.txt $ echo three &gt;&gt; test.txt $ echo four &gt;&gt; test.txt $ cat test.txt | sed '/w/ q' one two $   works as expected.  however when I try to tail the file  $ tail -f test.txt | sed '/w/ q' one two   it never exits.  tail does not stop even though the pipe is broken.  Does anybody know how to make tail exit when sed exits? If $AB = I$ then $BA = I$ If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.    I do not understand anything more than the following.   Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix.   If the entries of the matrix are not from a mathematical structure which supports commutativity, what can we say about this problem?  P.S.: Please avoid using the transpose and/or inverse of a matrix. Are logs kept for past at jobs? If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that would enable mathematicians to continue pursuing mathematics, should the unthinkable occur? What are the most accepted ways to visualize the results of an independent two sample t-test? Is a numeric table more often used or some sort of plot? The goal is for a casual observer to look at the figure and immediately see that they are probably from two different populations. A finite $p$-group cannot be simple unless it has order $p$ I am to trying to prove this theorem: A finite $p$-group cannot be simple unless it has order $p$.  I have this:   Let $G = P$ and $|G|=p$; then there exists $N$, a normal subgroup of $G$ by Lagrange's theorem, such that $|N| \mid |G|$ so, $|N|= |e| = 1$ or $|N| =  p$  If $|N|=p$ so $N=G$.  But I don't know how I can prove the part in the case of $|N|=|e| = 1$  Is my start correct?