This is a very simple fact to prove, but I am worried that my proof is so simple as to have no chance of being right.      For a vector space $V$, $-(-v) = v$ for any $v \in V$.    Proof. Let $v \in V$. By the additive inverse axiom,  $$v + (-v) = (-v) + v = 0,$$ so $v$ is the additive inverse of $-v$. Since the additive inverse in a vector space is unique, and we denote the a dditive inverse of $-v$ by $-(-v)$, we have $$v = -(-v).$$  Have I skipped any steps?
The exercise is: Prove that $-(-v)=v$ for every $v \in V$  Proof Suppose  $v \in V$ and $V$ is a vector space. Then $-(-v) \in V$ as result of the scalar multiplication property and $-(-v)=-(-1\cdot v)=-1 \cdot(-1\cdot v) =(-1 \cdot-1)\cdot v = 1 \cdot v = v $ The desired result $-(-v)=v$ holds.  The solution manual gives the proof:  Proof   I just wanted to make sure that the way I did the proof isn't missing anything?Thanks
If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints. Where to find my Favorites in Google Chrome? Where can I find my Favorites in Google Chrome ?  Where to find the folder that contains them ? PayPal API Listener Website Payments Standard URI RAID10 vs. RAID5 for data files We are trying to evaluate the potential performance improvements that could be gained by switching an existing applications .mdf files to a RAID10 LUN vs. the existing RAID5 LUN they are living on right now. The transaction log already lives on a RAID10 LUN.   So the question is, if we move data files to RAID10 would the end user see performance improvements in the GUI? or would it be just faster writes to the data files when a checkpoint occurs and the changes get written to the data file but no improvement would be seen at the end user level? How to deal with cross-thread access exceptions? I need to host several Apache virtual hosts with SSL from a single IP.  Now - I understand that because SSL wraps around the HTTP request, there's no way to know which host is being requested until a public key has been sent to the client first. This essentially breaks the possibility of SSL virtual hosts using a standard SSL certificate.  I have obtained a Unified Communications Certificate (UCC), otherwise known as a Subject Alternative Name (SAN) certificate. This allows me to serve the same certificate for multiple domains.  I would like this to be the certificate served by Apache for any SSL request - and then have Apache resolve the virtual host as usual, once the encryption has been established.  How should I configure Apache for this? I have tried to research how this can be done, but all I can find are quotes which say that it is possible, but no specifics:    wiki.apache.org/httpd/NameBasedSSLVHostsWithSNI     While Apache can renegotiate the SSL   connection later after seeing the   hostname in the request (and does),   that's too late to pick the right   server certificate to use to match the   request hostname during the initial   handshake, resulting in browser   warnings/errors about certificates   having the wrong hostname in them.   serverfault.com/questions/48334/apache-virtual-hosts-with-ssl     Incidentally, it is possible to have   multiple SSL-secured named virtual   hosts on a single IP address - I do it   on my website - but it produces all   sorts of warnings in the Apache logs,   and certificate warnings in the   browser. I certainly wouldn't   recommend it for a production site   that needs to look clean.   -David Jul 31 at 4:58   www.digicert.com/subject-alternative-name.htm     Virtual Host Multiple SSL sites on a single   IP address. Hosting multiple   SSL-enabled sites on a single server   typically requires a unique IP address   per site, but a certificate with   Subject Alternative Names can solve   this problem. Microsoft IIS 6 and   Apache are both able to Virtual Host   HTTPS sites using Unified   Communications SSL, also known as SAN   certificates.     Please help. I'm trying to find the name of a lower-budget science fiction film that was available for viewing on demands on MIT student cable in the late 1990s.  I suspect the film was made earlier in that decade, and it definitely seemed to have been influenced by RoboCop.  The plot involves characters is a somewhat dystopian near future.  One of the characters (named Hawkins, I think) is initially in a wheelchair, and that is interfering with his relationship with the woman he loves.  (He watches virtual reality simulations of himself having sex with her when he's alone at home.)  Maybe halfway through the movie, he decides there's no hope for their relationship, so he lets the bad guys turn him into a RoboCop-like cyborg.  I think he ends up fighting the other main characters, but I don't remember how the movie ends.  The film was clearly shot cheaply but competently.  The color palette was rather drab, I think, and there were few indoor sets.  It's possible that the film never got a normal release of any kind, since I don't know where the students at MIT cable got ahold of it.  So does anybody know what this film was? Should we sign-then-encrypt, or encrypt-then-sign? Is there a way to "hide" a normal windowed application in Windows?  I'm looking for something that removes it from the taskbar and possibly puts an icon in the system tray instead. I need to find out the modular inverse of 5(mod 11), I know the answer is 9 and got the following so far and don't understand how to than get the answer. I know how to get the answer for a larger one such as 27(mod 392) but am stuck because they are both low numbers.  11=5 (2)+1  5=1 (5) Is there a way in Adobe Creative Suite to "share palettes/swatches" across programs? I use a mixture of InDesign, Photoshop and Illustrator to produce the vast amount of printed media I design at my current position and I find myself faced with copy/pasting color values from one program to another almost all the time, I have tried to find a quicker way but have been unsuccessful. I use a program called  to copy and store my most used colors, but still, it's not the same as being able to select it from the swatches palette in Adobe CS.  So the main meat of the question is:   Can I share my colors/swatches between InDesign, Photoshop and Illustrator and if so, how? Why does Maximum Likelihood estimation maximize probability density instead of probability I am trying to understand Maximum likelihood estimation but it looks like I am missing something rather elementary.          suppose we have an iid random sample $X_1, X_2,..., X_n$ for which the probability density function of each $X_i$ is $f(x_i; \theta)$ where $\theta$ is an unknown parameter. Then, the joint probability density function of $X_1, X_2,..., X_n$ is given by:          $f(X_1=x_1,X_2=x_2,\cdots,X_n=x_n)=\prod_{i=1}^{n} f(x_i, \theta)$      In Maximum Likelihood estimation, we try to maximize $f$ as a function of $\theta$.  Question: Why do we maximize probability density instead of probability? In what way does it make sense?  Edit: As explained in the answers, the probability of choosing a finite number of sample points from a continuous probability distribution is zero, so maximizing probability doesn't make sense, but how does maximizing probability density make sense? Why are the shebang parts of the following elementary scripts not working:  $ cat hello.sh #! /bin/sh echo Hello $ ./hello.sh bash: ./hello.sh: /bin/sh^M: bad interpreter: No such file or directory  $ cat hello.py #! /usr/bin/env python3 print("Hello") $ ./hello.py : No such file or directory   whereas calling the interpreter manually is working:  $ sh hello.sh Hello $ python3 hello.py Hello Occasionally, I've come across a webpage that tries to pop open a new window (for user input, or something important), but the popup blocker prevents this from happening.  What methods can the calling window use to make sure the new window launched properly? How to move a file? Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure? Probably it is a weird thing I would like to achieve but: I would like to SSH to a remote host and then execute some bash commands like "alias" automatically to use them afterwards in an interactive mode. The remote host does not allow the personal environment.    Is this possible? Would "expect" do it? Or is there any other way to achieve this? Basically I would like to have my bashrc without modifying any files on the remote host. This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it? Password protecting a page I'm trying to password protect a page but it doesn't seem to be working. I've set the password for the page but when you navigate to it it just loads up like normal and doesn't ask for any passwords.  This is the loop I have on my page:  &lt;?php /**  * @package WordPress  * @subpackage Default_Theme  * Template Name: Page  */  get_header(); ?&gt;  &lt;?php global $post; if ( post_password_required( $post ) ) { ?&gt;  &lt;?php $fields = get_acf(); ?&gt;  &lt;script type="text/javascript"&gt;  (function() {   window.onload = function() {   ...  }} )  &lt;/script&gt;  &lt;div id="map"&gt;&lt;/div&gt;  &lt;?php } else { ?&gt;     Not passworded        &lt;?php if (have_posts()) : ?&gt;         &lt;?php while (have_posts()) : the_post(); ?&gt;          &lt;?php endwhile; ?&gt;     &lt;?php endif; ?&gt;         &lt;?      } // Page isn't password-protected     ?&gt; &lt;?php get_footer(); ?&gt;   I want to password protect everything within the page.  UPDATE:  I have custom content on the page so if possible I need to be able to wrap some code around it all.  Update: Cross tabulation of two categorical variables: recommended techniques