How to verify that $\mathbb R^\mathbb R$ is a direct sum of the Ue and Uo The $U_e$ and $U_o$ denote the set of all real-valued even/odd function on $\mathbb R$ respectively.
Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$ A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called even if $f(-x) = f(x)$ $ \forall x \in \mathbb{R}  $   A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called odd if $f(-x) = -f(x)$ $ \forall x \in \mathbb{R}  $   Let $U_{e}$ denote the set of real-valued even functions on $\mathbb{R}$ and let $U_{o}$ denote the set of real valued odd functions on $\mathbb{R}$.   Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$   This question from Q24 of "Linear Algebra Done Right" 3rd Edition - Chp 1.4 (Subspaces). The author of the book does minimal work with functions during the chapter. I have no idea how to approach the question.
How can I reset $PATH to its default value in Ubuntu? I accidentally unset all the directories of $PATH while trying to add a new one in ~/.bashrc. I opened a new terminal window as I was editing and now $PATH is empty. I'm worried if I boot from another drive to find the $PATH I won't be able to boot into this drive again.  Basically, what is the default result of echo $PATH? TexMaker (and TexShop) not working after installation of El Capitan How we can add two date intervals in PHP What is "%AppData%"? For example, "dude," "man," "buddy," "pal," etc, when used to stand in for someone's name.  "Hey, pal, how's it going?"  Is there a word for terms like these?  Or is "colloquialism" as close as we can get? How do you get JavaScript/jQuery Intellisense Working in Visual Studio 2008? I thought jQuery Intellisense was supposed to be improved with SP1. I even downloaded an annotated version of jQuery 1.2.6, but intellisense will not work in a separate jscript file. I have the jQuery library referenced first on my web page in the &lt;head> tag. Am I doing anything wrong? Why are there two different hats for "Where in the World"? I am wondering why some got a moon and others got a sun as Where in the World hat for the same action:   vs.   And is it possible two collect both e.g. from different sites? I only get moons…  I know that there has already been some spoilering . Case-insensitive attribute-value selector with Jquery Intuition about Herbrand Models. I'm having trouble understanding how Herbrand models work. I feel like there is a bigger picture to it which I'm not understanding. Why does every satisfiable first order formula have a herbrand model at all ? Can I somehow construct a Herbrand model from a normal model ? My book also says that : a formula is unsatisfiable iff the Herbrand expansion is unsatisfiable. The book I'm using &quot;Mathematical Logic for computer science by Mordechai Ben-Ari&quot; has all the proofs of the above but its not very intuitive to me. Could somebody share their point of view? I'm trying to do raster algebra with two rasters, but when I transform the rasters to array I can't do the operation because the files do not have the same extent  example:  from osgeo import gdal import numpy as np  IMG1 = gdal.Open('C:/IMG1.TIF') IMG2 = gdal.Open('C:/IMG2.TIF')  img1 = IMG1.ReadAsArray() img2 = IMG2.ReadAsArray()  img1+img2 &gt;&gt;&gt;&gt;&gt; ValueError: operands could not be broadcast together with shapes (5353,3352) (5548,3232)    So.. if I see the gt of every img show that img1 and img2 have the same size of the pixel (100) but different extent  g1 = IMG1.GetGeoTransform() gt1 &gt;&gt;&gt;&gt;&gt; (482084.999999999, 100.0, 0.0, 5652715.00001374, 0.0, -100.0) g2 = IMG2.GetGeoTransform() gt2 &gt;&gt;&gt;&gt;&gt; (475184.999999999, 100.0, 0.0, 5643715.00001535, 0.0, -100.0)   So.. I used this operation in order to get the intersect extent.. but I don't know how to apply  r1 = [gt1[0], gt1[3],gt1[0] + (gt1[1] * IMG1.RasterXSize), gt1[3] + (gt1[5] * IMG1.RasterYSize)] r2 = [gt2[0], gt2[3],gt2[0] + (gt2[1] * IMG2.RasterXSize), gt2[3] + (gt2[5] * IMG2.RasterYSize)] intersection = [max(r1[0], r2[0]), min(r1[1], r2[1]), min(r1[2], r2[2]), max(r1[3], r2[3])] I would change the font of section and subsection of my document. I have created it using book class. I would use this font:    According to Adobe pdf propriety, this font is AvantGarde-Demi. How can I use it in Latex. what command I should use to recreate this in latex. I think I should use titlesec package right? Profile changes in Q&A are not pushed to Area 51 I just noticed that the info text on  is an older version of the text in .  I found  from April 2016 where Adam Lear says:     Area 51 is a separate codebase. It's also heavily outdated. Currently we can push a profile update from Q&amp;A to Area 51, but not the other way around   I think the pushing isn't working anymore. I just updated my text on SO again and it was changed in all my accounts, except Area 51. Narration cartoon about a town that has weird paranormal occurrences in it How to toggle Scroll Lock in Excel 2010 without a Scroll Lock key in the keyboard I have a notebook with no Scroll Lock key (or any shortcut using the 'Fn' key).  Somehow Excel 2010 got the Scroll Lock on. I see it left down in the status bar and now I can't navigate through cells using the arrow keys, which is pretty annoying. It wasn't turned on some hours ago, so I think I must have hit some Scroll Lock toggle shortcut in Excel 2010 by mistake.  Anyone knows what hot to turn off Scroll Lock in Excel (shortcut or menu option)? How can I match a quote-delimited string with a regex? If I'm trying to match a quote-delimited string with a regex, which of the following is "better" (where "better" means both more efficient and less likely to do something unexpected):  /"[^"]+"/ # match quote, then everything that's not a quote, then a quote   or  /".+?"/   # match quote, then *anything* (non-greedy), then a quote   Assume for this question that empty strings (i.e. "") are not an issue. It seems to me (no regex newbie, but certainly no expert) that these will be equivalent.   Update: Upon reflection, I think changing the + characters to * will handle empty strings correctly anyway. I have an existing trigger which converts leads into peron accounts if the company name is blank.  Trigger AutoConvert on Lead (after insert) {      LeadStatus convertStatus = [           select MasterLabel           from LeadStatus           where IsConverted = true           limit 1      ];      List&lt;Database.LeadConvert&gt; leadConverts = new List&lt;Database.LeadConvert&gt;();       for (Lead lead: Trigger.new) {           if (!lead.isConverted &amp;&amp; lead.Company == null) {                Database.LeadConvert lc = new Database.LeadConvert();                String oppName = lead.Name;                 lc.setLeadId(lead.Id);                lc.setDoNotCreateOpportunity(true);                lc.setConvertedStatus(convertStatus.MasterLabel);                 leadConverts.add(lc);           }      }       if (!leadConverts.isEmpty()) {           List&lt;Database.LeadConvertResult&gt; lcr = Database.convertLead(leadConverts);      } }   I need to also write a trigger that stops records from coming into Salesforce if that email already exisits in a contact or an account (we use person accounts).  The issue is how do I get it working with the existing trigger (I need it to run before the trigger that converts it into a person account is run).   Is there a way to combine the two or add the functionality to create the record only if the email address doesn't already exist as a contact in the org? Thread Confinement How can I tell whether beetroot is still edible? I have some raw beetroot I bought about a week ago, having stored them at room temperature since. It's the first time I bought it raw, and I'm unsure whether it's still edible.   It is already soft, and when I cut it, it smells... Well, different than I expected. It's not a rotten smell as such, but it's nothing like the sweet beet-rooty smell I know from the pre-packaged, pre-boiled supermarket variety.  How can I tell whether it's still safe to eat? What are the prerequisites to studying general relativity? This question recently appeared :     Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade.   It seems like something that would be a good addition to this site: I think it's specific enough to be answerable but still generally useful. The textbook aspect is covered pretty well by , but beyond that: What college-level subjects in physics and math are prerequisites to studying general relativity in mathematical detail? How to tweet using terminal?