I know this is fairly trivial but how is $\dim\operatorname{range}T &gt; \dim W$? Basically, can someone explain the first inequality (as well as the second)?  $\mathbf{3.23}$ A map to a smaller dimensional space is not injective Suppose $V$ and $W$ are finite-dimensional vector spaces such that $\dim V&gt;\dim W$. Then no linear map from $V$ to $W$ is injective.  Proof$~$ Let $T\in\mathcal L(V,W)$. Then \begin{align*} \dim\operatorname{null} T  &amp;= \dim V-\dim\operatorname{range} T \\ &amp;\ge\dim V-\dim W \\ &amp;&gt;0\,, \end{align*}
I am trying to proof the theorem that states that:  "A map to a smaller dimensional space is not injective"  So I first suppose that V and W are finite-dimensional vector spaces such that dimV > dimW and I let T be the linear transformation from V to W.  Given the above, you have that:  dim(null(T)) = dim(V) - dim(range(T)) >= dim(V) - dim(W) > 0  So I understand the first and second equalities. However, I don't understand why dim(V) - dim(W) is strictly greater than 0.
How do I join a Windows Active Directory? We have a "mix-mode" Lab. and we want to migrate to Ubuntu, from Windows.  In this process, how do we add Ubuntu workstations to a Windows Domain / Active Directory ('AD') setup? Modern Linux commands that parallel uuencode/uudecode functionality I need to get binary files into text format.  Years ago, I used uuencode/uudecode, and it was easy:  uuencode file.xls &gt; file.txt # Delete file.xls # Do whatever with file.txt # Now I need file.xls back uudecode file.txt &gt; file.xls   But uuencode/uudecode is not on my system, and as I look around, the yum/rpm process is ugly--this gives me the clue that maybe no one uses it any more and instead there is something more modern to do this.  I looked at xxd and hexdump but couldn't get them to do it.  Could just be dumb user issues.  What can I use on Linux to do this?  Would like it to be something that tends to be in the native install so I don't need to yum/rpm it if possible.  Just for your own understanding, what I'm specifically trying to do now is source control an *.xls file but the corporate tool doesn't allow *.xls in.  I respect that constraint normally, but in my context, an *.xls is a configuration table for my code so It is logically like source code, but the corporate SVN trigger is physical--if it's *.xls, it gets rejected, so I'll encode it and source control the *.txt file.  Thanks! I have 14" 16:9 notebook and would like my Windows 10 taskbar have on the (left) side. Unfortunately, although I set it to small icons the bar is pretty wide. Do not you know how to make it thinner?  This is how I would like to have it: How can I extract the points from a ListSurfacePlot3D? Is there any way to verify a cashier cheque? There are plenty of warnings around about forged cashier's cheques, and that if you deposit a forged one your bank will "hold you responsible".   What does it mean that the bank will "hold you responsible"? Clearly the money won't be credited. Will you be charged a fee like an NSF cheque? Will they try to prosecute you? Is there any way to confirm whether a cashiers cheque is forged? Could you for example tell your bank that you were uncertain of the origin, and ask them to confirm it? Can you call the issuer and verify (from the serial number) that they did in fact issue a cheque for that amount with you as the payee? I'm trying to load data from different tables using left outer joins into a QgsVectorLayer. The approach I'm following so far is the one from the developers cookbook . The underlying database is postgres. The query returns correct data, including a geometry column, however I can't figure out how to get the join inside the query to work with . As the table the data come from needs to be specified, is it even possible to do so?  Would approaching the problem using psycopg2 be more suited, and if so, is there a way to create a layer from the result set? The only way I can think of is creating an empty layer and adding each feature individually by looping over the result set. But possibly there is an easier way?  Sadly, using the builtin DB Manager is not an option. The layer has to be added programmatically. Do spice containers need to be hermetic? Spice containers purchase  I'm planning on upgrading my spice storage. Reading some advice online (e.g. , and ) I came to the conclusion that spice containers should:   Not be exposed to direct sunlight and high temperatures Be airtight   Which means that either you store it inside of some closed cupboard/wardrobe/room or you have containers which do block light and are in a relatively shady place. And the containers need to be airtight.  Available solutions  I've been browsing for some containers available online in my country and the 2 most sensible options I've seen are these:       The problem  The first one is hermetic (it has some rubber around the lid) but it's clear glass. The other one does not have the rubber, just a regular twist-off lid, but it's amber glass.  It'll be hard for me to find any space inside of cupboards to store all of my spices (I need at least 30), so I'd rather buy amber glass container, which I believe will effectively block the sunlight, even when kept on the counter. But I can't find any hermetic amber-glass containers, only ones with regular lid.  The question  Is that regular twist-off lid enough to keep spices airtight and fresh? Or should I focus on buying a hermetic container (like the one in clear glass). I could then for example paint them black to block the sunlight. Bring Debian to second Position in the GRUB Menu at startup I've tried other solutions, but none of them are specific to my setup. I just started using the new Linux subsystem featured in the recent Windows 10 update. I'm using WAMP x64 v3.0.6, and have successfully cloned my website and got it up and running. At the root of the Linux subsystem, I installed Drush globally with Composer, using the directions given on the .  Typing drush status returns the following when I run it in my default directory:  Drupal version         :  7.50  Site URI               :  http://default  Database driver        :  mysql  Database hostname      :  localhost  Database port          :  Database username      :  root  Database name          :  aerosphere  PHP executable         :  /usr/bin/php  PHP configuration      :  /etc/php5/cli/php.ini  PHP OS                 :  Linux  Drush script           :  /home/bob/.composer/vendor/drush/drush/drush.php  Drush version          :  7.1.0  Drush temp directory   :  /tmp  Drush configuration    :  Drush alias files      :  Drupal root            :  /mnt/c/wamp64/www/drupal-7.50  Site path              :  sites/default   But, as mentioned in the title, drush cc all returns No Drupal site found, only 'drush' cache was cleared.[warning] I assume it has something to do with adding a path to the .bashrc, but I don't know what to add. Any tips would be greatly appreciated!  My question is different from "", they're on a Mac and using MAMP, and I'm using WAMP on Windows. I've tried the solutions offered on that post, but the issue has not been resolved.  I'm obviously having issues connecting to MySQL as I get this when I run mysql command:     The program 'mysql' can be found in the following packages: * mysql-client-core-5.5 * mariadb-client-core-5.5 * mysql-client-core-5.6 * percona-xtradb-cluster-client-5.5      Try: sudo apt-get install .   So I've added these lines to my .bashrc file:  PATH="$HOME/.composer/vendor/bin:$PATH" export PATH="/mnt/c/wamp64/mysql/mysql5.7.14/bin:$PATH"   The top line was added to get drush working. I've checked multiple times, and I have the path correctly set, but it's still not working. Can I tap a creature with summoning sickness to pay "tap an untapped creature you control" costs? I have a  on the battlefield and I played a vampire earlier this turn. Can I tap both to transform Chosen of Markov even though I just played the vampire?  Extra Example: I play  to put two 1/1 white bird tokens into play. Can I now tap a both of them and another creature I control to pay Battle Screech's flashback cost of tapping three white creatures?   Obviously these creatures couldn't use any tap abilities until my next turn, but since the tapping is coming from a different source, I figured the situation might be different. What is the use of the square brackets [] in sql statements? I've noticed that Visual Studio 2008 is placing square brackets around column names in sql. Do the brackets offer any advantage? When I hand code T-SQL I've never bothered with them.  Example:  Visual Studio:  SELECT [column1], [column2] etc...   My own way:  SELECT column1, column2 etc... Textbook on the Geometry of Special Relativity I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ?  I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do not need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth.  Thanks in advance. C programming: How to program for Unicode? How to calculate crc16 in php Guaranteed lifetime of temporary in C++? How to replace a video on YouTube I need to make a small correction in a video I uploaded to YouTube. Googling shows that this is not possible. However, I remember seeing it done before. I posted a comment on a video saying that a certain part of the video was missing. I got a reply back that it was fixed, and when I went to see the video again, it was changed. How was this done? Add Area 51 Meta to the app Where to place 'only' relative to prepositions? I know that questions about the placement of 'only', are often asked here; accordingly, I searched for an answer to my question before posting it.  Question  Where are focusing adverbs placed relative to prepositions?  For example,   For his presentation, he needed to talk only about tomatoes. For his presentation, he needed to talk about only tomatoes.   My Attempt to Answer the Question  In both examples, it seems to me that 'only' is placed close enough to what it is intended to modify (the topic of the subject's discourse: tomatoes) that I cannot see any justifiable misinterpretations.  One of the examples sounds better to me, but at this point in my studies, I am more interested in what the traditional rules of grammar require, than in how the wordings sound. Plotting discontinuous functions without spurious vertical segments Are there recommendations or pitfalls for playing an archer cleric? In my 4E game, a player is going to try an elven cleric with a longbow (and a mace, just in case). She chose mostly ranged powers. Are there any caveats to playing such a character, or any builds that I could recommend for her?