Confidence interval of linear regression variable calculation
Should confidence intervals for linear regression coefficients be based on the normal or $t$ distribution?
So i have this $3\times 3$ matrix  $$     \begin{pmatrix}     -2 &amp; 0 &amp; 0 \\     3 &amp; 1 &amp; -6 \\     0 &amp; 0 &amp; -2 \\     \end{pmatrix} $$  I want to check if the matrix is diagonizable. First thing i do is find the roots : namely $\lambda_1 = −2$ of multiplicity $2,$ and $\lambda_2 = 1$ of multiplicity $1.$  In order to ensure that the matrix is diagonalizable we need to find a basis of $\mathbb{R}^3$ whose elements are eigenvectors for $A.$ In particular, since we have the eigenvalue $\lambda_1=-2$ having multiplicity $2$ we need to check that the dimension of the eigenspace $E(\lambda_1)$ is equal to $2.$ We compute the eigenspaces of $A.$  $\DeclareMathOperator\Nul{Nul}$On the answer sheet is says that we need to find $\Nul(A+2Id)$... but why $2Id?$ Isn't is supposed to be $-2,$ so $\Nul(-2Id-A)$??   I do not understand this $\Nul(A+2Id).$ I always thought that we should compute $\Nul(-2Id-A),$ so what do i need to do ?   $\Nul(-2Id-A)$ gives me a different result from $\Nul(A+2Id).$  Does that mean that since $-2$ has a multiplicity of $2,$ we need to work with $2$ and not $-2$?  Somebody please help with a clear and simple explanation ! Thanks! Do I need a Japanese transit visa when arriving at Narita and departing from Haneda? Evaluation of Differentiation and Integration I was having some thought about what would be the syntax to make the end point evaluation of derivatives or integrals. Such as making the | with the two end points of evaluation on the top an bottom of the line. Any suggestions would be wonderful. How to connect to Salesforce REST API without OAuth? How does Jitterbit do it? How to connect to Oracle as "SYS" from SQL*Plus in Java Use mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes. Use  strong mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes.    I know the steps of weak mathematical induction... Basis step: $p(n)$ for $n=1$ or any arbitrary $n_0$ ... show that it is true Inductive hypothesis: $p(n)$ for $n=k$ ... Assume that it is true for $n=k$ Inductive step: $P(n)$ for $n=k+1$ ... Show that this is true for $n=k+1$ How to make itemize/enumerate/description environment robust to missing \item elements? Is there any way to make LaTeX tolerate empty itemize/enumerate (i.e. no \item)?  Specifically, I would welcome any package/elegant hack that would gracefully "cancel" a list if no elements are found (instead of firing up a Latex Error).  Before jumping to the obvious answer (which I expect would be something along the lines of "don't use itemize if you don't have items to put in" ;)) please consider my use-case:   I want to format remotely stored elements (say, bibliographic items), which I can only access as an XML file.  &lt;elems&gt;   &lt;bibelem&gt;&lt;title&gt;Cool result 1&lt;/title&gt;&lt;author&gt;joe&lt;/author&gt;&lt;/bibelem&gt;   &lt;bibelem&gt;&lt;title&gt;Cool result 2&lt;/title&gt;&lt;author&gt;jim&lt;/author&gt;&lt;/bibelem&gt; &lt;/elems&gt;  I have a python script that downloads/parses the XML and outputs its content in a file bib.tex as a list of Latex macros-decorated data  \BibElem{Cool result 1}{joe} \BibElem{Cool result 2}{jim}  Now, I want to grant the end-user complete control over the styling of her/his produced document, so the intended usage within Latex is something like  \documentclass{article} \begin{document}   \newcommand{\BibElem}[2]{\item {\em #1} by {#2}}   \begin{itemize}     \input{bib.tex}   \end{itemize} \end{document}    The rationale behind this design choice is that the user should be restricted as little as possible in her/his styling choices. For instance, the user may either want to display the bibelems within a list as shown in the example, or may prefer a fancy-formatted tabular. On the other hand, the user should design the latex document once, and should be able to run it regardless of the amount of data produced by the source.  Therefore, the python script cannot make any assumption regarding the context in which its latex product will be included, and it should be up to the user to deal with empty bibliographies, i.e. with empty lists. How could one possibly do that?  Update: Werner proposed a nice solution that redefines the list environment to make Latex oblivious of the fact he is currently in a list. However, this only seems to work if the empty list is the last list in the document, as revealed by the following mwa:    \documentclass{article} \makeatletter \newenvironment{myitemize}   {\itemize}   {\@newlistfalse\enditemize} \makeatother \begin{document}   % Empty Itemize   \begin{myitemize}   \end{myitemize}   % Next non-nested itemize environment (can be itemize/enumerate/description)    \begin{myitemize}     \item Arghh!   \end{myitemize} \end{document} SF Story about human adapting to alien environment How to use Alt+Shift to switch keyboard layouts? I recently upgraded to Ubuntu 13.10, with the new keyboard handling system. The little "switch keyboard" option does not let me set it to Alt+Shift. How can I get this keyboard binding, as I have absolutely no interest in using the Mac-esque Super+Space combo? Preloading images with jQuery I'm looking for a quick and easy way to preload images with JavaScript. I'm using jQuery if that's important.  I saw this here ():  function complexLoad(config, fileNames) {   for (var x = 0; x &lt; fileNames.length; x++) {     $("&lt;img&gt;").attr({       id: fileNames[x],       src: config.imgDir + fileNames[x] + config.imgFormat,       title: "The " + fileNames[x] + " nebula"     }).appendTo("#" + config.imgContainer).css({ display: "none" });   } };   But, it looks a bit over-the-top for what I want!  I know there are jQuery plugins out there that do this but they all seem a bit big (in size); I just need a quick, easy and short way of preloading images! TikZ: Bend text so that it follows a line I have two arrows going from a node on the left to a node on the right. Both arrows are slightly bent. I want to label the arrows with text that is bent just like the arrows. This is as far as I've got:  \begin{tikzpicture} \node (One) at (-3,0) [shape=circle,draw] {One};  \node (Two) at (3,0) [shape=circle,draw] {Two}; \draw [-&gt;, thick] (One) to [bend right=45]  (Two); \draw [-&gt;, thick] (One) to  [bend left=45] (Two); \node (mental) at (0,1.75) {Some bent text}; \node (non-mental) at (0,-1.75) {Some more bent text}; \end{tikzpicture} I'm not a git master, but I have been working with it for some time now, with several different projects.  In each project, I always git clone [repository] and from that point, can always git pull, so long as I don't have outstanding changes, of course.  Recently, I had to revert to a previous branch, and did so with git checkout 4f82a29.  When I was again ready to pull, I found that I had to set my branch back to master.  Now, I can not pull using a straight git pull but instead, have to specify git pull origin master, which is annoying, and indicates to me that I don't fully understand what is going on.  What has changed which does not allow me to do a straight git pull without specifying origin master, and how to I change it back?  UPDATE:  -bash-3.1$ cat config [core]     repositoryformatversion = 0     filemode = true     bare = false     logallrefupdates = true [branch "master"] [remote "origin"]     url = git@github.com:user/project.git     fetch = refs/heads/*:refs/remotes/origin/*   UPDATE 2: To be clear, I understand that my original method may have been incorrect, but I need to fix this repo so that I can simply use git pull again.  Currently, git pull results in:  -bash-3.1$ git pull You asked me to pull without telling me which branch you want to merge with, and 'branch.master.merge' in your configuration file does not tell me either.  Please name which branch you want to merge on the command line and try again (e.g. 'git pull  '). See git-pull(1) for details on the refspec.  If you often merge with the same branch, you may want to configure the following variables in your configuration file:      branch.master.remote =      branch.master.merge =      remote..url =      remote..fetch =   See git-config(1) for details.   I can tell git pull which branch to merge, and it works correctly, but git pull does not work as it did originally before my git checkout. Answering beyond the question's scope Is answering beyond the scope of the question a valid thing? Or should we just stick to the question asked and ignore any potential bugs which might be present (of course OP is unaware of that, as of now).  To make myself more clear, please have a look at this .  There are 3 answers which talk about the issue asked and that's it. If you look at my answer, I've added an additional point about a potential bug which the OP has missed now (may find out later). So is it a good thing to point out such potential bugs or not? Should this practice be encouraged? Find a quadratic equation with root $x_1^2$ and $x_2^2$ I'm struggling to solve this problem:  "If the equation $ax^2 + bx +c$ $(a \neq 0)$ admits real and not null roots of $x_1$ and $x_2$, obtain the equation which evaluates to roots $(x_1)^2$ and $(x_2)^2$"  I know the answer is $a^2x^2 -(b^2 - 2ac)x + c^2$ but I am not able to get to this answer, I was trying to derive it from the sum and product equations $S=-b/a$ and $P=c/a$ but the answer I'm getting to is a monstruous different thing.  Could someone explain to me how to get to this answer?  Thanks Why can't I run two mysqli queries? The second one fails What is the definition of a "feature map" (aka "activation map") in a convolutional neural network? Intro Background Within a convolutional neural network, we usually have a general structure / flow that looks like this:  input image (i.e. a 2D vector x)   (1st Convolutional layer (Conv1) starts here...)  convolve a set of filters (w1) along the 2D image (i.e. do the z1 = w1*x + b1 dot product multiplications), where z1 is 3D, and b1 is biases. apply an activation function (e.g. ReLu) to make z1 non-linear (e.g. a1 = ReLu(z1)), where a1 is 3D.   (2nd Convolutional layer  (Conv2) starts here...)  convolve a set of filters along the newly computed activations (i.e. do the z2 = w2*a1 + b2 dot product multiplications), where z2 is 3D, and and b2 is biases. apply an activation function (e.g. ReLu) to make z2 non-linear (e.g. a2 = ReLu(z2)), where a2 is 3D.    The Question The definition of the term &quot;feature map&quot; seems to vary from literature to literature. Concretely:  For the 1st convolutional layer, does &quot;feature map&quot; corresponds to the input vector x, or the output dot product z1, or the output activations a1, or the &quot;process&quot; converting x to a1, or something else? Similarly, for the 2nd convolutional layer, does &quot;feature map&quot; corresponds to the input activations a1, or the output dot product z2, or the output activation a2, or the &quot;process&quot; converting a1 to a2, or something else?  In addition, is it true that the term &quot;feature map&quot; is exactly the same as &quot;activation map&quot;? (or do they actually mean two different thing?)  Additional references: Snippets from :  *The nomenclature is being used loosely here. In particular, I'm using &quot;feature map&quot; to mean not the function computed by the convolutional layer, but rather the activation of the hidden neurons output from the layer. This kind of mild abuse of nomenclature is pretty common in the research literature.   Snippets from :  In this paper we introduce a visualization technique that reveals the input stimuli that excite individual feature maps at any layer in the model. [...] Our approach, by contrast, provides a non-parametric view of invariance, showing which patterns from the training set activate the feature map. [...] a local contrast operation that normalizes the responses across feature maps. [...] To examine a given convnet activation, we set all other activations in the layer to zero and pass the feature maps as input to the attached deconvnet layer. [...] The convnet uses relu non-linearities, which rectify the feature maps thus ensuring the feature maps are always positive. [...] The convnet uses learned filters to convolve the feature maps from the previous layer. [...] Fig. 6, these visualizations are accurate representations of the input pattern that stimulates the given feature map in the model [...] when the parts of the original input image corresponding to the pattern are occluded, we see a distinct drop in activity within the feature map. [...]  Remarks: also introduces the term &quot;feature map&quot; and &quot;rectified feature map&quot; in Fig 1.  Snippets from :  [...] One dangerous pitfall that can be easily noticed with this visualization is that some activation maps may be all zero for many different inputs, which can indicate dead filters, and can be a symptom of high learning rates [...] Typical-looking activations on the first CONV layer (left), and the 5th CONV layer (right) of a trained AlexNet looking at a picture of a cat. Every box shows an activation map corresponding to some filter. Notice that the activations are sparse (most values are zero, in this visualization shown in black) and mostly local.   Snippets from   [...] Every unique location on the input volume produces a number. After sliding the filter over all the locations, you will find out that what you’re left with is a 28 x 28 x 1 array of numbers, which we call an activation map or feature map. How do you join on the same table, twice, in mysql? I have 2 tables. One (domains) has domain ids, and domain names (dom_id, dom_url).  the other contains actual data, 2 of which columns require a TO and FROM domain names. So I have 2 columns rev_dom_from and rev_dom_for, both of which store the domain name id, from the domains table.  Simple.  Now I need to actually display both domain names on the webpage. I know how to display one or the other, via the LEFT JOIN domains ON reviews.rev_dom_for = domains.dom_url query, and then you echo out the dom_url, which would echo out the domain name in the rev_dom_for column.  But how would I make it echo out the 2nd domain name, in the dom_rev_from column? What is the C# Using block and why should I use it? What is the purpose of the Using block in C#? How is it different from a local variable? I have a following problem: when I plug-in my USB flash drive, it isn't shown in Windows Explorer or recognized by system at all. In device properties, it is shown as:     USB MEMORY BAR     There's no partitions on the drive, so I can't format it:    DISKPART:     Virtual Disk Service error:   There is no media in the device.   HP USB format tool:     There is no media in the specified device.   Any help will be greatly appreciated! Thank You! Is there a way to specify fragile mode as default for all frames? Now I know how to define a fragile frame: \begin{frame}[fragile].