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Here is a question as phrased in Advanced Linear Algebra (Steven Roman):     Show that a linear operator $\tau$ on a finite-dimensional complex inner product space $V$ is normal if and only if whenever $S$ is an invariant subspace under $\tau$, so is $S^\perp$.   I'm only interested in the $\Rightarrow$ direction. Wikipedia (and ProofWiki)  that uses some kind of computation with the trace. Linear Algebra Done Right chooses an orthonormal basis and then uses the fact that $\Vert \tau v \Vert = \Vert \tau^* v \Vert$.  However, the above  proofs seem a bit too computational to me. Can the following argument work?  (Use $\odot$ for orthogonal direct sum.) Suppose that $\tau$ is normal and write $$V=E_{\lambda_1} \odot \cdots \odot E_{\lambda_k}$$ where the $E_{\lambda_i}$ are the eigenspaces of $\tau$. If $S$ is a $\tau$-invariant subspace then $\tau|_S$ is diagonalizable, so $$S=\widetilde{E}_{\lambda_1} \odot\cdots\odot \widetilde{E}_{\lambda_k}$$ where the $\widetilde{E}_{\lambda_i} \le E_{\lambda_i}$ are the eigenspaces of $\tau|_S$ (which could be trivial). Then choose orthogonal complements $D_i$ such that $E_{\lambda_i} = \widetilde{E}_{\lambda_i} \odot D_i$, and then $$S^\perp = D_1 \odot\cdots\odot D_k.$$ Since each $D_i$ is a subspace of an eigenspace, they are all $\tau$-invariant. Let a,b,c,d be real or complex numbers with absolute value less than $1$. Is the following true:  $$|ab-cd| \leq |a-c| +|b-d|?$$ This is what I have come up with so far, but I am rather lost at lines 3-4...  $$\begin{align} (a,b)\in A\times (B\setminus C)\iff&amp; a\in A \land b\in (B\setminus C)\\ \iff&amp; a\in A \land (b\in B \land \lnot(b\in C))\\ \iff&amp; (a\in A \land b\in B) \land (a\in A \land b\notin C)\\ \iff&amp; (a\in A \land b\in B)\setminus (a\in A \land b\in C)\\ \iff&amp; (a,b)\in (A\times B)\setminus ((a,b)\in A\times C)\\ \iff&amp; (a,b)\in (A\times B)\setminus (A\times C) \end{align}$$ I'm on ub20LTS and I really dislike how much top space the topmost bars use. Any idea on how to remove or reduce their numbers?  Edit: Thanks to everybody's support, this is how my workspace looks like now. I've :  Installed gnome-extensions for firefox. Turned off menu bar in ff. Installed dash to dock. Installed hide top bar.  This way you can better utilize your screen space. How to really clear the terminal? I can issue the clear command or press Ctrl + L to clear the current Ubuntu terminal, but this just shifts previous output upwards and if you use mouse scroll or PgUP and PgDown keys it's hard to distinguish where the output of previous command ends and output of current command begins.  Is there a way to really clear the terminal so I won't see previous command results? My Prof wrote that   $\sum_{n=2}^{\infty} n(n-1)x^{(n-2)} = \frac{2}{(1-x)^3}$  when $-1&lt; x&lt;1$  I do not understand where the above formula comes from. Can someone please explain the steps or provide a hint?   Thanks! In Xubuntu 16.04.3 (and several previous versions aswell) it is not possible to have no power icon in the notification area.  A power icon is displayed even though 'Show system tray icon' is unchecked in Xfce Power Manager.  When 'Show system tray icon' is checked, there are two power icons displayed.  (See screenshots)  I've searched and I can't find any information related to this in Xubuntu/Xfce.  Is there a way to fix or work around this bug? Once a post has been edited, you can go into the revision list, and click "View Source".  If a post has not been edited (and I don't have the edit privilege) is there any way to see the raw Markdown?  The reason I ask I because of  - it's impossible to see the source markdown, and how it's different from the rendered Markdown.  Not exactly a huge issue, just curious! I found this heart while exploring the crimson biome - what is it? Irreducibles are prime in a UFD Does each new Horcrux take half of Voldemort's remaining soul, thus taking a smaller piece each time? When you commit a murder and create a Horcrux, you split your soul. It’s been talked a lot about Voldemort splitting his soul in 7/8 parts. But I think his remaining soul was not 1/7 or 1/8 but far more weaker than that because he did not create all Horcruxes at once.  When he created his first Horcrux, his soul was split into 50-50 (diary). But when the second Horcrux was created, it was created by Voldemort only and the first Horcrux soul should have remained intact. Hence it would have been a 25-25 split of Voldemort's main soul. And so on for every split.  Am I right? How did Rita know she lost the power? Wi-Fi not working on Lenovo ThinkPad E570 (Realtek RTL8821CE) We just bought a Lenovo  (which is supposed to have the Intel 8265 for wireless). This laptop was bought in part because it is certified to run Ubuntu and as such to me at least it would make sense for things to work out of the box... but no such luck.  First, I installed Ubuntu 17.10 and after installation realised Wi-Fi was not working. I also tried Ubuntu 16.04 from a USB drive since that is the version of Ubuntu it is certified for, but no luck there either.  The wireless chip is not detected by lshw (or even lspci by the looks of it), ip link does not show a wireless interface, and the iwlwifi kernel module is not loaded at boot (I can load this myself using modprobe iwlwifi but this does not make Wi-Fi work). I think it is a combined Wi-Fi/Bluetooth chip, and Bluetooth does appear to work (or at least Bluetooth shows up in Gnome and rfkill list).  modinfo iwlwifi | grep 8265 shows me a file ending in -34 is supposedly loaded but only a file ending in -33 (and some other lower numbers) is present in /lib/firmware, if that helps).  I have also installed the Linux 4.14.9 kernel in an attempt to get things to work, but this did not resolve the issue either.  Further information  root@ThinkPad-E570:~# modinfo iwlwifi | grep 8265 firmware:       iwlwifi-8265-34.ucode root@ThinkPad-E570:~# rfkill list all 0: tpacpi_bluetooth_sw: Bluetooth     Soft blocked: no     Hard blocked: no 1: hci0: Bluetooth     Soft blocked: no     Hard blocked: no root@ThinkPad-E570:~# dmesg | grep iwl root@ThinkPad-E570:~# lspci -nnk | grep 0280 -A3 05:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. Device [10ec:c821]     Subsystem: Lenovo Device [17aa:c024]   Update  I just noticed that the laptop does not have an Intel wireless chip at all, but in fact has a Realtek one... which means that the  of the shop I bought this laptop at were not accurate... Sigh. Same problem stands, however, just with a crappier wireless chip. Why do diamonds shine? I have always wondered why diamonds shine. Can anyone tell me why? Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$ . Find the value of $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+x^2} dx$     My attempt  First, I know the existence of this integral since $$ \mid \int_{-a}^{a} \frac{e^{-x^2}}{1+x^2} dx \mid \;\; \leq\;\;  \mid \int_{-a}^{a} \frac{1}{1+x^2} dx \mid$$  Next, let $R &gt;0 $ be an arbitrary and $\gamma_1(t) = -R(1-t) + Rt , \;\; 0 \leq t \leq1$ and $\gamma_2(t)= Re^{i\pi (t-1)}, \;\; 1 \leq t \leq 2 $.  So, take $f(z) = \frac{e^{-z^2}}{1+z^2}$.   I tried to find the value or upper bound of norm of $$\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz$$.  However, $$ \lvert\int_{\gamma_2} \frac{e^{-z^2}}{1+z^2} dz\rvert \leq  \frac{\text{max}\lvert e^{-z^2}\rvert}{R^2-1}\pi R = \frac{e^{R^2}}{R^2-1}\pi R $$ So, I can't proceed this.  May I ask you how to solve this? I'm trying to compute this integral, $$\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx \hbox{ where } 0&lt;a&lt;1$$ I drew a typical Pacman contour with branch cut at positive real axis. Then, we have $$\int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz=\left(\int_{L_{1}}+\int_{L_{2}}+\int_{C_{R}}+\int_{C_{\epsilon}}\right)\frac{\ln z}{z^{a}(z+1)}dz$$ The function has a pole at $z=-1$. Then, by residue theorem, the whole integral becomes  \begin{align*} \int_{\Gamma}\frac{\ln z}{z^{a}(z+1)}dz&amp;=2\pi i\text{res}(\frac{\ln z}{z^{a}(z+1)};-1)\\&amp;=2\pi i\lim_{z\rightarrow-1}(z+1)\frac{\ln z}{z^{a}(z+1)}\\&amp;=2\pi i\lim_{z\rightarrow-1}\frac{\ln z}{z^{a}}\\&amp;=2\pi i\frac{\ln e^{i\pi}}{e^{i\pi a}}\\&amp;=2\pi i\frac{\ln1+i\pi}{e^{i\pi a}}\\&amp;=-\frac{2\pi^{2}}{(e^{i\pi})^{a}} \end{align*} On $L_1$, $z=xe^{i\epsilon}$ where $\epsilon\leq x\leq R$  On $L_{2}$, $z=xe^{i(2\pi-\epsilon)}$ where $\epsilon\leq x\leq R$  On $C_{R}$, $z=Re^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  Lastly on On $C_{\epsilon}$, $z=\epsilon e^{i\theta}$ where $\epsilon\leq\theta\leq2\pi-\epsilon$  So I tried to compute them separately and I anticipate the curves will both go to $0$ by ML inequality, and two linear integrals will consist of the term $\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx$ and hence simply solve for it. But I'm having trouble physically solving it out. I think the log is throwing me off. Can anybody please help me? I  want to download free app in my iPod touch device and i have the following doubts.  Kindly resolve me doubts.   when i try to download it asks my credit card security number. since i am trying to download free app , why do i need to provide my credit card security code? Will i be debited for downloading the free app from app store?   I cant able to find any information regarding this on apple.  Please provide me some references about this from apple and please resolve my doubts.  Thanks a lot. When is it okay to exceed the absolute maximum rating on a part? Run bash script at login stored in the home folder? When I try to load a LaunchAgent plist from launchctl I can't find out how to run a script in the home directory.  My code is:  &lt;?xml version="1.0" encoding="UTF-8"?&gt; &lt;!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd"&gt; &lt;plist version="1.0"&gt; &lt;dict&gt;     &lt;key&gt;ProgramArguments&lt;/key&gt;     &lt;array&gt;         &lt;string&gt;bash&lt;/string&gt;         &lt;string&gt;~/script.sh&lt;/string&gt;     &lt;/array&gt;     &lt;key&gt;RunAtLoad&lt;/key&gt;     &lt;true/&gt;     &lt;key&gt;Label&lt;/key&gt;     &lt;string&gt;com.tyilo.test&lt;/string&gt; &lt;/dict&gt; &lt;/plist&gt;   I have tried both with and without bash and also replacing ~ with $HOME. I have also tried using bash -c without it working.  The error code is:   `com.tyilo.test: bash: ~/script.sh: No such file or directory` Using Squeeze Theorem to show the following limits Consider a function f (x) defined on R satisfying:   for all x distinct 0.  Calculate: lim f(x) with x→∞ and limf(x) with x→−∞   Issue information:   if x → ∞ then we can consider | x | = x  if x → -∞ then we can consider | x | = -x  I can use x² = | x |?