Computing correlation coefficient in linear regression I'm working with a linear regression model given by the following formula $$y = \beta_1x + \beta_0 + \epsilon.$$ I am to prove that the correlation coefficient between $\hat{y}$ and $\epsilon$ is equal to zero if the model is correct. Basically this problem comes down to prove that $Cov(\hat{y}, \epsilon) = 0$. I'm a bit stuck here. I would appreciate any help.
Understanding linear regression In class we've seen that $a$ (the weights) must satisfy $$X^T (y-Xa) =0$$  Here $X$ is a $(n\times d)$ matrix (so we have $n$ samples in $\mathbb R^d$)  let's denote the residuals $r = y-Xa$. In our class notes, it is written that     The normal equations require the residuals to be orthogonal to each of the columns of $X$.   Why?     Therefore, the solution of the linear regression is a projection of $y$ onto the subspace spanned by $v_1 , \ldots , v_d$ (the columns of $X$)   Can you please explain this?
Ideal of the twisted cubic The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute $I(X)$, which equals by the Nullstellensatz the radical of the ideal $I := (ad-bc,b^2-ac,c^2-bd) \subseteq k[a,b,c,d]$. I think that that $I$ is already a radical ideal, even a prime ideal. Namely, I suspect that  $$\phi^* : Q:=k[a,b,c,d]/I \to k[s,t] , a \mapsto s^3, b \mapsto s^2 t , c \mapsto s t^2 , d \mapsto t^3$$ is an injection. If this is true: How can we prove that? I've already tried to find a $k$-basis of the quotient, but this turned out to be a big mess. Even the representation of the quotient as a monoid algebra doesn't seem to help. Another idea is the following: A formal manipulation of generators and relations implies $Q_a \cong k[a,b]_a$. Thus it suffices to prove that $Q \to Q_a$ is injective, i.e. that $a$ is not a zero divisor. How to print every Nth character of a string? This line of code does exactly this except I want it to start at the first character not the 7th. $ echo 1234567890abcdefghijklmnopqrstuvwxy | sed 's/.\{1,7\}\(.\{0,1\}\)/\1/g'  What I'm looking for is 19gow Adverb position in "Listen carefully to what I say" I've just installed ubuntu server 17.10. During installation, it suggested me to connect to the network through wifi, and so i did. After installation was finished and system was rebooted, computer connected to the wifi automatically (which means, it saved the connection creds somewhere). But in /etc/network/interfaces i found nothing. I need my computer connect to wifi network with static ip, so I've put configuration into the /etc/network/interfaces:  # /etc/network/interfaces -- configuration file for ifup(8), ifdown(8) # Generated by debian-installer.  # The loopback interface auto lo iface lo inet loopback   iface wlp2s0b1 inet static         wpa-driver wext         address 192.168.0.12         netmask 255.255.255.0         gateway 192.168.0.1         wpa-ssid *****         wpa-psk **********         dns-nameservers 8.8.8.8 192.168.0.1 auto wlp2s0b1   When system starts, it doesn't connect using my configuration. It does only after  sudo ifdown wlp2s0b1 &amp;&amp; sudo ifup -v wlp2s0b1  and after this command, system getting a SECOND IP! Server is still available by the ip it gets from DHCP, and in the same time, it is available by the static ip! Clicking to reload an updated answer removes the code snippet buttons and removes syntax highlighting Between vs. Among - "Difference [between | among] one thing and other things." Consider the following two versions of a question:1   Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms?  Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms?    Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited"at the center.  And Version 2 might subtly suggest a more free-form comparison like the Mesh or Fully Connected, with the target sentence simply "one among many":   Between approx equal "Star" and among approx equal "Fully Connected"?  Don't get me wrong: I do think this is quite subtle, not always applicable, and I don't think anyone would intuit such a "network topology" meaning without further explanation.   So the main question is, are both grammatically correct?  If yes, is there a semantic difference to choose one over another, perhaps as I suggested?  Or does the plurality aspect of between and among trump other semantic and stylistic issues?    1. In my question , another user provided a very helpful edit. However, the change of the word between to among prompted this question. This question is pretty simple: I'd like to have the possibility to get the exact size of a picture in LaTeX with, for example, the macro \the.  In fact, I've founded the way \settoheight{} and so on, but it didn't work at all. What I've got doesn't match with what I can see in the image properties (I suppose that's the name in English). The result: 8pt versus 1300 pixels. It's may be a mis-use...  Edit: In the end, my aim was to find a way to fit the picture in LaTeX. If width and height were both less than the dimensions of the body, then it should just be centred. It look like this :     Require:   \usepackage{graphicx} \usepackage{xifthen} \usepackage{calc}   Code (pretty ugly, I agree):   % Centrering+fitting for image, with captation and label (not auto) \newlength\imageheight \newlength\imagewidth \newcommand{\pic}[3]{% \settoheight\imageheight{\includegraphics{pictures/#1.png}}% \settowidth\imagewidth{\includegraphics{pictures/#1.png}}% \ifthenelse{\lengthtest{\imagewidth &gt; \textwidth}}{\ifthenelse{\lengthtest{\imageheight &gt; \imagewidth}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics[height=0.85\textheight]{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics[width=0.95\textwidth]{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}}{\ifthenelse{\lengthtest{\imageheight &gt; \textheight}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics[height=0.85\textheight]{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}{\par\vspace{0.95em}% \begin{figure}     \centering\includegraphics{pictures/#1.png}     \caption{#2}     \label{#3} \end{figure}% \par\vspace{0.95em}}}} %   To use it : put your png files in a "pictures" directory, then, just simply call the picture like this : \pic{name_whitout_extension}{Captation_if_needed}{Label_if_needed}.    Edit 2 :   Thanks to Martin Scharrer's easyfig package, I've got a more or less 1,6 times faster solution to put my pictures, centred, with the perfect size.  CTAN:   VC:   Version v1.2 – 2012/05/15    This could be useful for some others, I hope. Pretty printing XML with javascript I have a string that represents a non indented XML that I would like to pretty-print. For example:  &lt;root&gt;&lt;node/&gt;&lt;/root&gt;   should become:  &lt;root&gt;   &lt;node/&gt; &lt;/root&gt;   Syntax highlighting is not a requirement. To tackle the problem I first transform the XML to add carriage returns and white spaces and then use a  tag to output the XML. To add new lines and white spaces I wrote the following function:   function formatXml(xml) {     var formatted = '';     var reg = /(&gt;)(&lt;)(\/*)/g;     xml = xml.replace(reg, '$1\r\n$2$3');     var pad = 0;     jQuery.each(xml.split('\r\n'), function(index, node) {         var indent = 0;         if (node.match( /.+&lt;\/\w[^&gt;]*&gt;$/ )) {             indent = 0;         } else if (node.match( /^&lt;\/\w/ )) {             if (pad != 0) {                 pad -= 1;             }         } else if (node.match( /^&lt;\w[^&gt;]*[^\/]&gt;.*$/ )) {             indent = 1;         } else {             indent = 0;         }          var padding = '';         for (var i = 0; i &lt; pad; i++) {             padding += '  ';         }          formatted += padding + node + '\r\n';         pad += indent;     });      return formatted; }   I then call the function like this:  jQuery('pre.formatted-xml').text(formatXml('&lt;root&gt;&lt;node1/&gt;&lt;/root&gt;'));   This works perfectly fine for me but while I was writing the previous function I thought that there must be a better way. So my question is do you know of any better way given an XML string to pretty-print it in an html page? Any javascript frameworks and/or plugins that could do the job are welcome. My only requirement is this to be done on the client side. Challenges with Key with Gaps? I know that there's a difference between relativistic and rest mass. Relativistic mass is "acquired" when an object is moving at speeds comparable to the speed of light. Rest mass is the inherent mass that something has regardless of the speed its moving at.  When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$.   How does the rest mass of an object turn into energy? Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles.  Also, Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy? Title, section headings disappears when I use \usepackage{forest} with IEEE Access template When I include a  \usepackage{forest} all colored text title, section headings disappears with IEEE Access journal template How can I tackle this problem? On SO, when I'm scrolling through "questions" some have a yellow background.  Is this intuitive to everyone but me?  What is the yellow background trying to tell me about this particular question? Is there a simple, constructive, 1-1 mapping between the reals and the irrationals? Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?  I know that the Cantor–Bernstein–Schroeder theorem implies the existence of a 1-1 mapping between the reals and the irrationals, but the proofs of this theorem are nonconstructive.  I wondered if a simple (not involving an infinite set of mappings) constructive (so the mapping is straightforwardly specified) mapping existed.  I have considered things like mapping the rationals to the rationals plus a fixed irrational, but then I could not figure out how to prevent an infinite (possible uncountably infinite) regression. Short Question:  Why can we manipulate a read-only file in Vim using : + w + q + ! even without being an administrator?  Long Question:  I have a text file (myFile.txt) which is read-only for everyone:  navid@navid-ThinkPad-T530:~/ubuntuTest$ ls -l myFile.txt  -r--r--r-- 1 navid navid 26 Aug 22 21:21 myFile.txt   I can open it with Vim without having admin privileges:  navid@navid-ThinkPad-T530:~/ubuntuTest$ vi myFile.txt    I modify it and press: Esc + : + w + q + Enter and I see this error message:  E45: 'readonly' option is set (add ! to override)   So far, everything makes sense.  But when I press: Esc + : + w + q + ! + Enter, Vim saves the changes.  I'm using Ubuntu 16.04 and VIM 7.4. filling area between 2 functions, with shading Why didn't the machine affect Bond in Spectre? What is the measure of the set of numbers in $[0,1]$ whose decimal expansions do not contain $5$? How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...  Thanks in advance... Show inline math as if it were display math (and vice versa) How to view monitor refresh rate in macOS? I just purchased a Dell P2175Q monitor for my 15" MBP w/Touchbar. I'm connecting with a USB-C to DisplayPort cable, but when I go to Apple > System Preference > Displays, there is nothing about the Refresh Rate. Pressing the Option key does not show any additional options.  I've seen countless screenshots where this information should appear. Is there another way to check the refresh rate?  I'm on macOS 10.13.4  Thanks.