I don't know where to start on this proof. I know that a linearly independent set is a set where c1v1+c2v2+⋯+cnvn=0 has no non-trivial solutions other than c1=c2=...=cn=0.   How can I use that to figure out this proof?
Is the above statement true? If I have already proven that the set is linearly independent, is it automatically given that the subset is also linearly independent? If so, how would I go about showing that as a proof.
There are numerous (, , ) questions concerning this problem. I realize this is a borderline duplicate, but I'll re-iterate the futility of this particular filter and why it should be outright removed from Stack Overflow.  Do you know how effective this is in preventing users who just don't care from posting questions with bad titles? 100% ineffective. What do these users do?   Replace it with a synonym: "issue," "trouble," "predicament," "botheration" or what have you. Or worse, they simply work around the filter by deliberately misspelling the word or inserting space or punctuation:    () ?    and the coup de grâce:      I mean, even Jeff . This adds zero value to the title. In fact, it looks worse. Why, oh why are people  to scrap a bad feature? All this does is:   Prevent people who have a  to use the word "problem" in the title from posting questions with an appropriate title. Break the search feature because of workarounds or the need to use less descriptive titles than the error itself   This is an extraordinarily useless (and harmful) feature. Please, for the love of Cthulhu,  remove, eliminate, abolish, eradicate, extirpate this "feature" from Stack Overflow.    Bonus:     So I can't mention the Halting Problem in the title? – Keith Thompson Nov 1 at 19:05      @KeithThompson Maybe it's a "Halting Issue". Or "Need help with Halting". Or the "Halting thing in Computability Theory". – slhck Nov 1 at 19:14 How to change ubuntu install from legacy to uefi How do I disable a bottom panel with the list of open windows on GNOME 3? Many sysv init scripts used a corresponding file in /etc/default to allow the administrator to configure it. Upstart jobs can be modified using .override files. How do I override or configure systemd units, now that systemd is the default in Ubuntu? I'd like to colorize a cell based on its contents. 0 = red, 100 = green, and linearly interpolated between.  function LinInt(x){       var ss = SpreadsheetApp.getActiveSheet();   var cell = ss.getActiveRange();   var hue;   hue = (x/100)*120;    var color = HSVtoRGB(hue, 40, 100);   cell.setBackground(color);    return x;     }     function HSVtoRGB(h, s, v) {     var r, g, b, i, f, p, q, t;     if (h &amp;&amp; s === undefined &amp;&amp; v === undefined) {         s = h.s, v = h.v, h = h.h;     }     i = Math.floor(h * 6);     f = h * 6 - i;     p = v * (1 - s);     q = v * (1 - f * s);     t = v * (1 - (1 - f) * s);     switch (i % 6) {         case 0: r = v, g = t, b = p; break;         case 1: r = q, g = v, b = p; break;         case 2: r = p, g = v, b = t; break;         case 3: r = p, g = q, b = v; break;         case 4: r = t, g = p, b = v; break;         case 5: r = v, g = p, b = q; break;     }   return '#'+((b | g &lt;&lt; 8 | r &lt;&lt; 16) / 0x1000000).toString(16).substring(2); }   When I do this I get an error saying that I am not allowed to call setBackground. I assume this is because I'm not allowed to change the color of one cell from another because I'm screwing up getting the current cell that the function is running in.  How do I get the cell that the function is running in so that I can call setBackground() on it?  For example, I'd like to type =LinInt(50) into cell A1 and have A1 be yellow and have the number 50 in it. Then when I type =LinInt(100) into cell A2 it is green and has the number 100 in it. I'm trying to use itemize inside a tabularx environment, but I have a problem when I mention the end of the line. It seems that using \\ end the line after the end of itemize, resulting in a supplementary vertical spacing that I would like to remove.  Any hint?  \begin{frame} \frametitle{Sujet de thèse}         \begin{center}             \textit{Sujet N$^\circ$221 : Formation et dynamique des champs de dunes longitudinales.}         \end{center}          \begin{tabularx}{\textwidth}{l !{\color{blue}\vline width 3pt} X}             Contexte &amp;  \scriptsize{Classification des dunes selon leur mécanisme de croissance, leur orientation et leur forme.              Bonne compréhension :             \begin{itemize}[noitemsep,topsep=0pt, label=\textbullet]                 \item du mécanisme de croissance lié à une disponibilité en sédiment grande.                 \item de la croissance sous un vent unidirectionnel.             \end{itemize}} \\             Objectifs &amp; \scriptsize{Etude de l'orientation longitudinale :             \begin{itemize}[noitemsep,topsep=0pt, label=\textbullet, parsep=0pt ]                 \setlength\itemsep{0cm}                 \item Développement d'un formalisme sous des vents multidirectionnels.                 \item Etude des dunes longitudinales dans le cas d'une faible disponibilité en sédiment. (Fingering mode)                 \item Inversion: données topographiques $\longrightarrow$ Données de vents $\rightarrow$ paléo-climats.              \end{itemize}} \\             Outils &amp; \scriptsize{Modèle numérique (Rescal), Données expérimentales, Théorie.} \\         \end{tabularx}     \end{frame}     And yes, dunno if it's important but everything is inside beamer. Execute a script upon logout/reboot/shutdown in Ubuntu I'd like to run a .sh file using bash on logout and shutdown of Ubuntu. I have searched up and down the web and have found only fixes for KDE and GNOME. The script is simple, I just need to know if I can add the line of code to the "logoff/shutdown" file or if I need to reference it in another script. After upgrading to 12.10 quantal, the packaged version of fglrx no longer works. I discovered that this is because there is a separate 'legacy' fglrx driver for the HD 2k-4k series cards, but it is incompatible with the xorg server on 12.10.  This is the most current version of the driver for HD 2000 through HD 4000 series cards. You can't use the non-legacy fglrx driver, but you can use the open-source radeon driver if you prefer your WM compositing to be laggy and your YouTube videos to play like they would on a Pentium MMX series:    Usually this driver can be installed in the following way, necessary because apt-get install fglrx would pull in the non-legacy driver:  wget http://www2.ati.com/drivers/legacy/amd-driver-installer-12.6-legacy-x86.x86_64.zip unzip amd-driver-installer-* sudo sh ./amd-driver-installer-*.run --buildpkg Ubuntu/quantal sudo dpkg -i fglrx*.deb sudo aticonfig --initial -f   If you use a different version of fglrx (for example, a newer 12.9 that doesn't support those cards) then the final command will give you an error no supported hardware detected or something similar. However, everything works at this point and you will get a reasonable xorg.conf:  ... other stuff  Section "Device"         Identifier  "aticonfig-Device[0]-0"         Driver      "fglrx"         BusID       "PCI:1:5:0" EndSection  ... other stuff   At this point you're supposed to reboot and everything will be working with the fglrx driver. However, upon rebooting, you'll be treated to the following errors in Xorg.0.log when fglrx attempts to load:  (EE) Failed to load /usr/lib/xorg/modules/drivers/fglrx_drv.so: /usr/lib/xorg/modules/drivers/fglrx_drv.so: undefined symbol: noXFree86DRIExtension   Some searching around will show that this is a problem with the legacy ATI drivers not supporting xserver 1.13 or newer. () ATI has released a fixed driver for its most recent (HD 5000 series or later) cards, but not for the 'legacy' cards yet. The non-legacy ATI drivers can't be used with the old cards.  What should an Ubuntu user, using one of these HD 2000-4000 series cards, do?    Wait for an updated 'legacy' ATI driver that properly works with xserver 1.13?   Downgrade back to 12.04 Precise, which uses xserver 1.11? Try to downgrade xserver on 12.10 Quantal to 1.12, which could possibly break Unity and GNOME? Forced upgrade to HD 5000 series or later card? (Not possible with integrated graphics...) Some other 1337 action that fixes this problem painlessly? Unable to change payment information to NONE in iOS Appstore How does the momentum operator act on state kets? I have been going through some problems in Sakurai's Modern QM and at one point have to calculate $\langle \alpha|\hat{p}|\alpha\rangle$ where all we know about the state $|\alpha\rangle$ is that $$\langle x|\alpha\rangle=f(x)$$ for some known function $f$. ($|\alpha\rangle$ is a Gaussian wave packet.) Sakurai says that this is given by:  $$\langle p\rangle = \int\limits_{-\infty}^{+\infty}\langle\alpha|x\rangle\left(-i\hbar\frac{\partial}{\partial x}\right)\langle x|\alpha\rangle dx.$$  I am wondering how we get to this expression. I know that we can express   $$|\alpha\rangle =\int dx|x\rangle\langle x|\alpha\rangle$$  and  $$\langle\alpha|=\int dx\langle\alpha|x\rangle\langle x|,$$   so my thinking is that we have:  $$\langle\alpha|\hat{p}|\alpha\rangle =\iint dx dx'\langle\alpha|x\rangle\langle x|\hat{p}|x'\rangle \langle x'|\alpha\rangle, $$  and if we can 'commute' $|x\rangle$ and $\hat{p}$ this would become: $$\iint dxdx'\langle\alpha |x\rangle\hat{p}\langle x|x'\rangle \langle x'|\alpha\rangle,$$  which is the desired result as $$\langle x|x'\rangle=\delta(x-x').$$ Is this approach valid?  I think my question boils down to: Does the operator $\hat{p}$ act on the basis kets $|x\rangle$ or on their coefficients? In the latter case, if we had some state $|\psi\rangle = |x_0\rangle$ for some position $x_0$, then would we say that for this state $$\langle p\rangle =\langle x_0|\left(-i\hbar\frac{\partial}{\partial x}\right)|x_0\rangle = 0$$ as the single coefficient is $1$ and the derivative of $1$ is $0$? It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.   I'd like to do that but I am afraid I need help,   I think the first thing I need to figure out is how a general element in the commutator subgroup looks like, so that I can check that the defining condition for normality is satisfied.  That is, supposing for a moment that a general element in $G^c$ is denoted by $g$, I need to show that $aga^{-1} \in G^c,$ for all $a \in G$.  But here I get stuck, first because I am unsure how to write a general element in $G^c$ - a simple product in $G^c$ is of the form $xyx^{-1}y^{-1}aba^{-1}b^{-1}$ where $a,b \in G$. I cannot see a way to simplify this - I am sure there is one, but somehow I am blind today.  The second thing then is, even if one tries out the conjugation of a simple element like $xyx^{-1}y^{-1}$ in $G^c$, again not simplification offers itself easily I think .. what am I missing ?  An alternative would be to find a homomorphism of $G$ whose kernel is precisely $G^c$ - here I tried to think of this as a map $G \times G \to G$ but whatever I cook up is not a homomorphism.  Thanks for your hints !! How do I use large bitmaps in .NET? I'm trying to write a light-weight image viewing application. However, there are system memory limitations with .NET.  When trying to load large bitmaps (9000 x 9000 px or larger, 24-bit), I get a System.OutOfMemoryException. This is on a Windows 2000 PC with 2GB of RAM (of which 1.3GB is used up). It also takes a lot of time to attempt loading the files.  The following code generates this error:  Image image = new Bitmap(filename); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Point(0, 0)); }   As does this code:  Stream stream = (Stream)File.OpenRead(filename); Image image = Image.FromStream(stream, false, false); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Rectangle(0, 0, 100, 100), 4000, 4000, 100, 100, GraphicsUnit.Pixel); }   Also, it is enough to do just this:  Bitmap bitmap = new Bitmap(filename); IntPtr handle = bitmap.GetHbitmap();   The latter code was intended for use with GDI. While researching this, I found out that this is in fact a memory issue where .NET tries to allocate twice as much as is needed in a single contigous block of memory.    I know from other applications (Internet Explorer, MS Paint etc.) that it IS possible to open large images, and rather quickly. My question is, how do I use large bitmaps with .NET?  Is there anyway to stream them, or non-memory load them? I remember reading a sci-fi story where astronauts visit a planet, which is covered in stationary aliens with mile long necks that filter feed from the atmosphere.  It was published between 1960 and 2000.    It was in English and involved a pair of astronauts that went from planet to planet on a tour or something, though I don't know details of any other planets they visited. How do I import a namespace in Razor View Page? Get content from a url using php Why Changing Permissions For Some Systems Files is Not Allowed? How does a circular lens produce rectangular shots? Since Ubuntu 11.10 use a new kernel, it's very difficult to have a decent support for virtualization.   VirtualBox doesn't support guest additions for ubuntu 11.10, so I can't copy to and from my ubuntu desktop and windows, which I absolutely require, plus FreeBSD seems not to be able to use DHCP without guest additions.  Virt-manager instead gives an error on launch:  Unable to open a connection to the libvirt management daemon.  Libvirt URI is: qemu:///system  Verify that:  - The 'libvirt-bin' package is installed  - The 'libvirtd' daemon has been started  - You are member of the 'libvirtd' group   unable to connect to '/var/run/libvirt/libvirt-sock', libvirtd may need to be started: Permission denied  Traceback (most recent call last):   File "/usr/share/virt-manager/virtManager/connection.py", line 1146, in _open_thread     self.vmm = self._try_open()   File "/usr/share/virt-manager/virtManager/connection.py", line 1130, in _try_open     flags)   File "/usr/lib/python2.7/dist-packages/libvirt.py", line 102, in openAuth     if ret is None:raise libvirtError('virConnectOpenAuth() failed') libvirtError: unable to connect to '/var/run/libvirt/libvirt-sock', libvirtd may need to be started: Permission denied   The problem is solved by running virt-manager as root, but I don't like that. How do I change permissions to run Virt-Manager as user?  Is there a way to install guest additions on Ubuntu 11.10? I know that VARIABLE=value creates an environment variable, and export VARIABLE=value makes it available to processes created by the current shell. env shows the current environment variables, but where do they live? What comprises an environment variable (or an environment, for that matter)? How to make a domain receiving email behind cloudflare?