If a linear map, $φ: U→V$ is bijective, then $φ(x)=0$ has only the solution $x=0$.
Prove: If a linear map, $φ: U→V$ is bijective, then $φ(x)=0$ has only the solution $x=0$.
Word for person who looks for flattery What word could be used to describe a person who seeks and seems to depend on constant positive feedback from the people around him. This  shows the dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$ is $K^{*} = \{(x,t) \mid  \| \boldsymbol{x} \|_{\infty} \le t\}$. Is it right? How is it proved? Pluralizing abbreviations where the noun is not the last word How to create a generic array in Java? Visualforce Date formatting Add table row in jQuery systemd gives us the systemctl commands suite which is mostly used to enable services to start at boot time. We can also start, stop, reload, restart and check status of services with the help of systemctl.  We can do, for example, sudo systemctl enable service_name, and service_name will automatically start at boot time. We can also disable services not to start at boot time.  Is the only difference between the service and systemctl commands that systemctl can be used to enable the start of services at run time? Can we use systemctl on any service? What other significant differences are there? How do I set up my studio for shooting large rugs from the ceiling? How can I check if an application is running from a batch (well cmd) file?  I need to not launch another instance if a program is already running. (I can't change the app to make it single instance only.)  Also the application could be running as any user. Rename command not found when running a script I'm trying to to execute a script, which uses the command rename.   When I execute it, I get a message rename: command not found. Any ideas?   if [[ -z '$3' ]]; then     shopt -s dotglob     rename "$EXPRESSION" ./* else     rename "$EXPRESSION" "$3" fi What is the shortcut for opening a new window (opening folders). for that I have to go to folder options and right click and then need to click on open new window.  or I have to make focus on the folder icon or open a folder then press  Ctrl+N  for opening a new window is there any direct command spcific to open a new window in ubuntu. Counterexample for the normalizer being a normal subgroup CHAOS members have (had?) :     Because they are doing so many edits at once, CHAOS operates in a special mode where editing a title does not bump it to the home page.    I moderate at Writers.SE, and I can attest that this ability would be extremely beneficial to mods as well, and for the same reasons. On beta sites, wide-ranging changes are not infrequent. An established tag might be ruled problematic; a certain class of questions might need cleanup; etc. etc.. When this happens, the ensuing edits flood the front page with old content (frequently the somewhat problematic content the new change has come to address...). Furthermore, since beta sites are much less active than established ones, this flood can drown out recent activity, and then wallow around on the front page for some time.  I propose, then, the implementation of a moderator tool for editing without bumping. The value to beta sites is clear; the danger of abuse, as a mod tool, is nonexistent. Ubuntu 12.04 gnome issue, no clock is shown, strange starter icon cannot be removed i am finding nowhere a solution I am running UBUNTU 12.04 and using GNOME desktop, which I installed additionally. Logging into classic GNOME I see the following:     WHAT is this "earth globe" icon coming from?Seems to be named "starter". It is configured to open a website, from my bookmarks. WHY is it not showing the TIME?   Using dconf-editor I had no luck in answering the above questions.  Any idea? How do I remove the proprietary ATI drivers? I've installed 11.10 and the proprietary ATI drivers using "additional drivers" The performance of my system is absolutely awful and it shouldn't be. I tried to remove the proprietary drivers using the Additional Drivers tool and it appears to remove them. However after I reboot I cant get back into my desktop properly (the panel and launcher go missing). This doesn't seem to be an isolated problem in 11.XX.  covers how to restore the desktop (panel and launcher), but the guide doesn't fix my problem though.   Whenever I do sudo unity --reset it runs through its normal processes until it hangs at setting update "run_key" and never gets past that. I must reinstall the proprietary drivers using jockey-text or jockey-gtk in order to get back to my proper desktop.   Interestingly enough the system performance seems improved while it is in its "broken" state (missing panel and launcher).  I think restoring the default drivers may solve my problems but I cant figure out how to do it. Comma before which I'm not sure which of the following sentences is correct. Is the comma mandatory, optional or unnecessary? I've googled a bit about this topic, but I got confused.     Our mapping contains 2000 words, which map to more than one lemma.   or     Our mapping contains 2000 words which map to more than one lemma.   Context: In total, there are more than 2000 words.  I've read  . If I understand the answers correctly, my "which map to more than one lemma" is a defining clause and no comma is required. But is it wrong to add one? I wanted to left align a set of equations. I've tried the solution provided in  . It worked. But, I wanted to have a single equation number for the entire flalign in order to refer to it later. Like in case of equation+split. While using equation+align, I am able to get it. But, if I use equation+flalign the equations are still being centered. Is there a way to do it?  Example using equation+align,  \begin{equation}      \begin{align}       \mbox{Minimize } &amp; x_1+x_2+x_3 \\       \mbox{Subject to} &amp; \\        &amp; x_1+x_2 \leq 10 \\        &amp; x_2+x_3 \leq 8 \\        &amp; x_1+x_3 \leq 5       \end{align} \end{equation}   I would like to do this while the equations are left aligned.  It will be even better if the equations can be spaced a little (for example, 1 cm) from the left margin instead  of starting from the left margin.  Thanks in advance Can more than one antibody bind the same antigen I'd like to clear something up about antibodies that I'm not sure I've understood in the articles I've read. Looking at concepts such as "affinity maturation", "monovalent antigens" and "polyvalent antigens" it seems as if there are multiple antibodies which can bind to the same antigen. (Although I think I'm right in saying that individual antibodies can only bind to one antigen?) However, when I read sentences like "It has been estimated that humans generate about 10 billion different antibodies, each capable of binding a distinct epitope of an antigen" on the Wikipedia page (for antibody) it does make it sound like it's one antibody per antigen? Is this the case? Or is it one antibody per epitope. Either way, to put this question in a nutshell:  Is it true to say that the same substance could bind with (and, therefore, be recognised by) more than one antibody, or not?  Also, if the answer is no, does that mean that antibodies with different binding affinities are classed as the same antibody or not? I'm writing an article on  and some people pointed out that this is completely useless.  My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.  I came with some ideas like the Turing machine but I think I'm not grasping the right examples.  Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'? How to install HPLIP on my Ubuntu to support my HP printer and/or scanner?