I wrote a proof for the above question, but I am not sure whether it is right or not since I assumed linear independence.   Here's the proof:   Let $u$,$v$ be linearly independent vectors in $V$. $span(u)$, $span(v)$, $span(u+v)$ are all $T-invariant$.   $T(v)$ is an element in $span(v) \Longrightarrow T(v) = av$  $T(u)$ is an element in $span(u) \Longrightarrow T(u) = bu$  $T(u+v)$ is an element in $span(u+v) \Longrightarrow T(u+v) = T(u)+T(v)= bu+av = c(u+v)$  Hence, $bu+av-c(u+v) = 0 \Longrightarrow (b-c)u + (a-c)v = 0$ Since u and v are linearly independent, $(b-c)=(a-c)=0 \Longrightarrow b=a$   Hence, $T(w) = kw$, for all $w$ in $V$, and $k$ any scalar.   Is it right to assume linear independence? And is there any problem with my proof?
Let $T: V \rightarrow V$ be a linear operator.  I need to demonstrate that if all nonzero vectors of $V$ are eigenvectors of $T$, then there is one specific $\lambda \in K$ such that $T(v) = \lambda v$, for all $v \in V$.  I understand that, if all nonzero vectors of $V$ are eigenvectors of $T$, then $T$ must be a scaling transformation. It just stretch or shrinks vectors, but doesn't change their  directions.  So, the statement says that if it happens, then, there is a single $\lambda$ such that $T(v) = \lambda v$. In other words, if there is such transformation, then it scales all vectors by the same scalar $\lambda$.  Applying the transformation to our standard basis vectors, we have: $$ T(e_1) = \lambda_1 e_1 \\ T(e_2) = \lambda_2 e_2 \\ \vdots \\ T(e_n) = \lambda_n e_n $$  I understand I need to prove that $\lambda_1 = \lambda_2 = \dots = \lambda_n$, but I can't see how!  EDIT  $$ v = c_1e_1 + c_2e_2 + \dots + c_ne_n \\ T(v) = \mu v = \lambda_1c_1e_1 + \lambda_2c_2e_2 + \dots + \lambda_nc_ne_n \\ $$  Since what's multiplying $v$ coordinates is $\lambda_i$, then all of them must be $\mu$. I'm not sure how to 'mathematize' this. Is this idea correct?  EDIT 2 Extending the left hand side of EDIT 1, we have: $$ \mu v = \lambda_1c_1e_1 + \dots + \lambda_nc_ne_n \\ \mu(c_1e_1 + \dots + c_ne_n) = \lambda_1c_1e_1 + \dots + \lambda_nc_ne_n \\ \mu c_1e_1 + \dots + \mu c_ne_n = \lambda_1c_1e_1 + \dots + \lambda_nc_ne_n \\ $$  And since $e_i$ are linearly independent, $\mu = \lambda_1 = \lambda_2 = \dots = \lambda_n$. Is this proof correct?
What are "Community Wiki" posts? Pass ALL Arguments from Bash Script to Another Command As of today, a  has been found affecting versions 1.0.1 through 1.0.1f (inclusive) and 1.0.2-beta.  Since Ubuntu 12.04, we are all vulnerable to this bug. In order to patch this vulnerability, affected users should update to OpenSSL 1.0.1g.  How can every affected user apply this update now? Show (within ZF Set Theory without choice) that for any sets $A,B,C:$   If $|A|^{|B|} = |B|$, then $|A| = 1 = |B|.$   That is, if there is a bijection from $|A|^{|B|}$ to $|B|$, then  $|A| = 1 = |B|.$  Please help me.. How to shift EXIF date/time created by time in days, hours, minutes? Genetically modified humans study insectoid alien miners with super-intelligent hive-mind This is a short story that appeared in a collection. Two genetically-modified humans (corporate anthropologists, basically) travel to a planetoid inhabited by an insect race that is mining the planet in order to study the race. While there, they are practically ignored, as the individual members of the race lack consciousness. The humans determine a type of pheromone is used to control the worker caste, and co-opt several members to create a living area. They make plans to put the mining towards human ends, but before that can happen they are discovered. The presence of the artificial pheromones wakes up a hive-mind, super-intelligent bug, who then bargains with the surviving member of the team (the man) after killing and taking over the female member of the team.  The collection was read at some point between 2000-2005. I have been invited to write a book chapter in an edited volume, relating to my field of work. I have a paper ready that I have not published and was planning to submit it soon. So, I am wondering is it worth to use that article for the book chapter, as both are on the same topic, or is it in a sense a waste of a journal paper, as book chapters might not be seen to have the same academic value? What should I do? Should I, instead, send the ready paper to a journal for a review and write a book chapter, based on a couple papers I have published, for instance?  Generally speaking, where does a book chapter stand? Is it something between a conference paper and a full-fledged journal article? Or, does does it have the same value as either of the above-mentioned publication types, in your opinion? Finding the limit of roots products $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$ Office 2016 Advanced Setup I just started the Office 2016 Setup/Installation on my Windows 10 PC. The installation worked fine and started automatically showing only this dialog:    But there are no additional settings possible! I need to choose which programs I want to install and also add additional tools that are often missing when doing a default installation.  How can I change the Office Installation Settings?   Choose programs to install Add/Remove Office Tools (Maybe) Office installation directory   When I check after the installation in the the windows settings, the Office can only be uninstalled - no modifications available: Radius of Convergence when using Series[] I'd like to know how I can recursively (edit: iteratively) compute variance, so that I may calculate the standard deviation of a very large dataset in javascript. The input is a sorted array of positive integers. I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? I can't get to the bottom of this error, because when the debugger is attached, it does not seem to occur.  Collection was modified; enumeration operation may not execute  Below is the code. This is a WCF server in a Windows service. The method NotifySubscribers() is called by the service whenever there is a data event (at random intervals, but not very often - about 800 times per day). When a Windows Forms client subscribes, the subscriber ID is added to the subscribers dictionary, and when the client unsubscribes, it is deleted from the dictionary. The error happens when (or after) a client unsubscribes. It appears that the next time the NotifySubscribers() method is called, the foreach() loop fails with the error in the subject line. The method writes the error into the application log as shown in the code below. When a debugger is attached and a client unsubscribes, the code executes fine. Do you see a problem with this code? Do I need to make the dictionary thread-safe? [ServiceBehavior(InstanceContextMode=InstanceContextMode.Single)] public class SubscriptionServer : ISubscriptionServer {     private static IDictionary&lt;Guid, Subscriber&gt; subscribers;      public SubscriptionServer()     {                     subscribers = new Dictionary&lt;Guid, Subscriber&gt;();     }      public void NotifySubscribers(DataRecord sr)     {         foreach(Subscriber s in subscribers.Values)         {             try             {                 s.Callback.SignalData(sr);             }             catch (Exception e)             {                 DCS.WriteToApplicationLog(e.Message,                    System.Diagnostics.EventLogEntryType.Error);                  UnsubscribeEvent(s.ClientId);             }         }     }          public Guid SubscribeEvent(string clientDescription)     {         Subscriber subscriber = new Subscriber();         subscriber.Callback = OperationContext.Current.                 GetCallbackChannel&lt;IDCSCallback&gt;();          subscribers.Add(subscriber.ClientId, subscriber);                  return subscriber.ClientId;     }      public void UnsubscribeEvent(Guid clientId)     {         try         {             subscribers.Remove(clientId);         }         catch(Exception e)         {             System.Diagnostics.Debug.WriteLine(&quot;Unsubscribe Error &quot; +                      e.Message);         }     } } How can I reuse a Common Table Expression Connect MacBook Pro to two external monitors Is there a way to connect my MacBook Pro to two external monitors (either VGA or HDMI; no Thunderbolt) in such a way that the Mac screen serves as a third monitor?  What hardware do I need? Should smilies be turned into images in questions and answers? A lot of mention has been made of the seriousness of Stack Overflow (and presumably Server Fault). Would it be a good idea to turn  :-) into actual images as per  in questions and answers? The Android OS monitors the power consumption of running apps and hardware (display). As you may guess, this is a useful feature as it allows to track down energy greedy apps.  Is there an application that would do such monitoring on Mac OS?  It should calculate the percentages of the charge that an application/process/hardware has used.  I use plenty of little software that runs in the background permanently. It would be nice to know if such little helpers drain the battery significantly. Reason of existense of the extra space in align environment and removal of it between two align environments This is a follow-up question regarding the extra vertical space in amsmath's align environment. Why when in use align uses so much vertical space in contrast with other environments like for example tabular or itemize?  \documentclass[12pt]{article} \usepackage[top=0.3in, bottom=1.2in, left=0.8in, right=0.8in]{geometry}   \usepackage[fleqn]{amsmath} \usepackage{unicode-math}  \setlength{\mathindent}{0cm} \newcommand{\3}{\vspace{0.3cm}}  \title{} \author{} \date{}  \begin{document}  70  \begin{align*} &amp;e^{jz}=\cos z+j\sin z\\ &amp;\cos z=(1/2)(2\cos z)=\\ &amp;=(1/2)(2\cos z+j\sin z-j\sin z)=\\ &amp;=(1/2)(\cos z+j\sin z+\cos z-j\sin z)=\\ &amp;=(1/2)(e^{jz}+e^{-jz}) \end{align*}  80  \begin{itemize} \item Item 1  \item Item 2  \end{itemize}  90  \begin{tabular}{c|c}  Entry 1&amp; Entry 2  \\  Entry 3&amp; Entry 4 \end{tabular}  \end{document}     Also another image of the huge space between two align environments. Even without the blank lines the space is a lot.    \begin{document}  \begin{multicols*}{2}   \begin{align*} &amp;\text{70}\\ &amp;e^{jz}=\cos z+j\sin z\\ &amp;\cos z=(1/2)(2\cos z)=\\ &amp;=(1/2)(2\cos z+j\sin z-j\sin z)=\\ &amp;=(1/2)(\cos z+j\sin z+\cos z-j\sin z)=\\ &amp;=(1/2)(e^{jz}+e^{-jz}) \end{align*}    \begin{align*} &amp;\text{207}\\ &amp;x(t)=A_{c}\cos \theta(t)\\ &amp;\theta(t)=2\pi f_{c}t+\phi(t)\\ &amp;\phi(t)=K_{p}m(t)\\ &amp;x(t)=A_{c}\cos[2\pi f_{c}t+K_{p}m(t)] \end{align*}   Edit I:  As I mention in a comment I have tried the following code and there is no difference between two align environments, only between text and an align environment. Why doesn't work in the case of the two environments and how can I fix it so to work? Also what is the reason that this space exists?  \setlength{\abovedisplayskip}{0pt} \setlength{\belowdisplayskip}{0pt} \setlength{\abovedisplayshortskip}{0pt} \setlength{\belowdisplayshortskip}{0pt} "wait more than" vs. "wait FOR more than" If you make him wait (for) more than ten minutes, he'll just walk away.  I found both instances on Google Books:  and . So I'm a bid confused. Delete all files except files with the extension pdf in a directory I have a directory that contains the following:  x.pdf y.zip z.mp3 a.pdf   I want to delete all files apart from x.pdf and a.pdf. How do I do this from the terminal? There are no subdirectories so no need for any recursion.