Let $W$ be a vector space and let $U$ and $V$ be finite dimensional subspaces. Not sure how to go about solving this.
Let $U,W$ be subspaces of a vector space $V$. Show that   $$\dim(U)+\dim(W)=\dim(U+W)+\dim(U\cap W)$$   Hint: Show that the map given by $L:U×W\to V$ given by $L(u,w)=u-w$ is linear.   I can show that $L:U×W\to V$ given by $L(u,w)=u-w$ is a linear map. I also know that the dimension of $U×W$ is $\dim(U)+\dim(W)$. What do I do next? Any hints?
I want to make a plane for my supply on my PCB and a plane for my MCU to isolate the MCU from the power supply noise.  I found this article in Altium designer :  But it isn't clear how the split plane tool work. Can someone explain it in a different manner?  [EDIT]  This is a picture how I see the planes: How do you select the fastest mirror from the command line? I want to update my sources.list file with the fastest server from the command line in a fresh Ubuntu Server install. I know this is trivially easy with the GUI, but there doesn't seem to be a simple way to do it from from the command line? What's the difference between Wubi and a regular "alongside Windows" installation? Audio is working, but the speaker test doesn't work I've installed Ubuntu 12.04 LTS from a minimal cd on a netbook (Asus 1001 PXD). I've installed the ubuntu-desktop package using the --no-install-recommends option.  Everything works fine, except the "sound test" for headphones or analog speakers.  Clicking on the "test" buttons (front left and front right) I don't hear any sound. Despite this, the audio is working properly.  I've checked the audio levels with alsamixer; I have also checked that the test sounds actually exist in /usr/share/sounds/alsa.  I tried an installation of Ubuntu 12.04 LTS made with the desktop-cd, and in this case the speaker test works properly.  I suppose, therefore, that the problem could depend on the lack of a package, but I have not identified which one. chpasswd apply changes in the chroot directory What problem does oversampling, undersampling, and SMOTE solve? My phone was just stolen an hour or so ago. Are there any options for tracking? I realize this is a long shot, but my phone was lost or stolen a few hours ago.  At the time it was stolen, I didn't have any specific applications installed to track it or wipe the data. So... my bad. I'm hoping the fact that I have a pattern lock will at least prevent anyone from accessing my data.  My question is, do I have any options for tracking or data wipe after the fact? (I'm more interested in tracking.)  Note: I have read similar questions, but unless I'm mistaken, they all address the question in terms of preparation before losing the phone. I'm asking about fallback options after losing the phone. How to use hook_node_presave to change field's value I would like to upload a file asynchronously with jQuery.     $(document).ready(function () {     $("#uploadbutton").click(function () {         var filename = $("#file").val();          $.ajax({             type: "POST",             url: "addFile.do",             enctype: 'multipart/form-data',             data: {                 file: filename             },             success: function () {                 alert("Data Uploaded: ");             }         });     }); }); &lt;script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.2.0/jquery.min.js"&gt;&lt;/script&gt; &lt;span&gt;File&lt;/span&gt; &lt;input type="file" id="file" name="file" size="10"/&gt; &lt;input id="uploadbutton" type="button" value="Upload"/&gt;     Instead of the file being uploaded, I am only getting the filename. What can I do to fix this problem? Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$ In PHP, how do you change the key of an array element? I have an associative array in the form key =&gt; value where key is a numerical value, however it is not a sequential numerical value. The key is actually an ID number and the value is a count. This is fine for most instances, however I want a function that gets the human-readable name of the array and uses that for the key, without changing the value.  I didn't see a function that does this, but I'm assuming I need to provide the old key and new key (both of which I have) and transform the array. Is there an efficient way of doing this? Is it possible to paginate posts correctly that are random ordered? I found this problem on Wordpress Support and the topic unfortunately is now closed.  I have this same issue... (read below)    We have created a site where members can recommend things like favourite books, movie, songs, etc. For this problem I will use the Movies page as an example.  The "Movies" page is ultimately a custom page template that asks wordpress to display a random list of ALL posts that have been given the category "movies" (category 31). It displays the title of these movies in random order using the code below.  &lt;?php  $rand = new WP_Query("cat=31&amp;showposts=-1&amp;orderby=rand");  while($rand-&gt;have_posts()) : $rand-&gt;the_post(); ?&gt;" rel="bookmark" title="Permanent Link to &lt;?php the_title_attribute(); ?&gt;"&gt;&lt;?php the_title(); ?&gt; &lt;?php endwhile; ?&gt;   Problem is the list is getting rather long and I would like to split it up into two or more pages of about 10 movies each. To achieve this, I have used the code below.  &lt;?php  $page = (get_query_var('paged')) ? get_query_var('paged') : 1;  query_posts("cat=31&amp;orderby=rand&amp;showposts=10&amp;paged=$page");  while ( have_posts() ) : the_post()  ?&gt;" rel="bookmark" title="Permanent Link to &lt;?php the_title_attribute(); ?&gt;"&gt;&lt;?php the_title(); ?&gt; &lt;?php endwhile; ?&gt;   But there is a problem because although it splits up the data into pages of 10 posts each (paginates), it fails to include a new set of 10 posts on page 2 and so forth. In other words, because it is listing things in a random order, it just goes out and gets another 10 random posts (or in this case movie titles). As a result, we have some repeated movie title posts instead of a new set of 10 random movie titles on page 2, etc.  My question is - what can I add to this code to get wordpress to "remember" which random 10 posts it included on page 1, and then have it get a new set of 10 posts to put on pages 2, 3, etc. until all posts are displayed. I would like there to be only one occurrence of a post per page when sorted randomly in sets of 10. Book wherein humans were most dangerous/aggressive of numerous species I recall there were two warring alien empires, one of which was led by a race that could telepathically inflict pain on others, but when they tried to do it to humans, it would backfire. something about human's animal instincts being much closer to the surface than the other species.  Humans made the most aggressive and dangerous fighters, so both empires tried to reruit humanity, since the other races involved in the war were fairly monolithic and sided all with one empire or the other. Humanity was all splintered though.  The empire led by the telepathic race claimed that everyone in it was equal but in reality the telepathic race was in charge. Can I backup my entire Ubuntu installation to ISO DVD? How can I prove $\sqrt{2} ^{\sqrt{2}}$ is irrational? Indian CGPA to US GPA system Creating queries to select by attribute with arcpy I'm quite new with arcpy and I'm finding selecting by attribute a bit of a nightmare. There's a problem with my query on the script below [query = "\'Location\' = observer"], but I can't tell what it is. 'Location' is a field of the feature class I'm dealing with (fc_copy).  with arcpy.da.UpdateCursor(fc_copy,field_names) as cursor:        for row in cursor:         observer = row[0]         arcpy.MakeFeatureLayer_management(fc_copy, 'fl_copy_OBS')         arcpy.MakeFeatureLayer_management(fc_copy, 'fl_copy_TAR')         query = "\'Location\' = observer"         arcpy.SelectLayerByAttribute_management('fl_copy_OBS', 'NEW_SELECTION', query)         arcpy.SelectLayerByAttribute_management('fl_copy_TAR', 'NEW_SELECTION', query, invert_where_clause = 'INVERT')   This is the error I get => Invalid expression                            Failed to execute (SelectLayerByAttribute).  Is there any documentation to learn the rules to follow to build queries for arcpy? this works in viewport but not in render.  import bpy from math import *  def frame_handler(scene):     z = scene.objects['Lift'].location[2]     scene.objects['Text'].data.body = str(floor(z / 4.0) - 2)  bpy.app.handlers.frame_change_pre.clear() bpy.app.handlers.frame_change_pre.append(frame_handler)   How to get it working in render? I've read  and changed from "bpy.data.objects" to "scene.objects" as already seen in the script. What am I missing? Blender v. 2.81.16  Edit. Tried that depsgraph thing mentioned in comments and changed to frame_change_post:  import bpy from bpy.app.handlers import persistent from math import *  @persistent  # maybe not relevant but doesn't harm anything? def frame_handler(scene, depsgraph):     z = scene.objects['Lift'].evaluated_get(depsgraph).location[2]     scene.objects['Text'].data.body = str(floor(z / 4.0) - 2)  bpy.app.handlers.frame_change_pre.clear() bpy.app.handlers.frame_change_post.clear() bpy.app.handlers.frame_change_post.append(frame_handler)   Now the Text doesn't show up at all in rendered (but works in viewport). Windows of an application have minimum size and I can't find the window anymore I had an application (in case it matters: LibreOffice Calc] in full-size mode. I then clicked the middle of the three buttons of the top right to adjust the size. When I did that, the window became minimum size, literally a vertical line. I panicked and alt-tabbed, and now I can't even find the vertical line anymore. Opening new windows in that application opens all of them hidden to me, and closing/reopening or even restarting the OS doesn't help. I don't know how to resize my invisible windows anymore. Here's the preview of the windows in the application launcher: a vertical line (that I don't find on my screen anywhere if I click this). Prove that a function is not injective