Bijectivity of T(z) = λ z + μ z* Let $T = \lambda z + \mu \bar z$ where $\lambda, \mu \in \mathbb{C}$. Show that $T$ is bijective exactly when $\lambda \bar\lambda \ne \mu \bar\mu$.
How can I find such a $z$? (Injectivity of $T(z) = \lambda z + \mu\bar{z}$) We consider $T:\mathbb{C}\to\mathbb{C}$ defined by $T(z)=\lambda z+\mu\overline{z}$, where $\lambda ,\mu\in\mathbb{C}$. I want to prove that if $T$ is injective, then $\lambda\cdot\overline{\lambda}\neq\mu\cdot\overline{\mu}$.  Clearly, if $\mu =0$, then $\lambda\neq 0$ because $T$ is injective.  Now, if $\lambda\cdot\overline{\lambda}=\mu\cdot\overline{\mu}$ and $\mu\neq 0$, then I wanted to find a $z\in\mathbb{C}\setminus\{0\}$ such that $z=-\dfrac{\overline{\lambda}}{\overline{\mu}}\overline{z}$, because we would have that:  $T(z)=-\lambda\dfrac{\overline{\lambda}}{\overline{\mu}}\overline{z}+\mu\overline{z}=\overline{z}(-\lambda\dfrac{\overline{\lambda}}{\overline{\mu}}+\mu)=0$  ('Cause this would prove that $\lambda\cdot\overline{\lambda}=\mu\cdot\overline{\mu}$ implies $T$ is not injective.)  So, the question is, can we always find such a $z\in\mathbb{C}\setminus\{0\}$?
When is the limit of a sum equal to the sum of limits? I was trying to solve a problem and got stuck at the following step:  Suppose ${n \to \infty}$ .  $$\lim \limits_{n \to \infty} \frac{n^3}{n^3} = 1$$  Let us rewrite $n^3=n \cdot n^2$ as $n^2 + n^2 + n^2 + n^2 \dots +n^2$,$\space$ n times.  Now we have  $$\lim \limits_{n \to \infty} \frac{n^3}{n^3} = \frac {n^2 + n^2 + n^2 + n^2 + n^2 \dots +n^2}{n^3} $$  As far as I understand, we can always rewrite the limit of a sum as the sum of limits ...  $$\dots = \lim \limits_{n \to \infty} \left(\frac{n^2}{n^3} + \frac{n^2}{n^3} + \dots + \frac{n^2}{n^3}\right)$$  ...but we can only let ${n \to \infty}$ and calculate the limit if all of the individual limits are of defined form (is this correct?). That would be the case here, so we have:  $= \dots \lim \limits_{n \to \infty} \left(\frac{1}{n} + \frac{1}{n} + \dots + \frac{1}{n}\right) =$[ letting ${n \to \infty}]$ $= 0 + 0 + \dots + 0 = 0$  and the results we get are not the same.  Where did I go wrong? proof of functional completeness of logical operators If I know that the set of operators {∨, &amp; , ¬} is functionally complete, how do I go about proving/disproving the functional completeness of the following set of operators?  a) $\{\vee,\neg\}$  b) $\{\to,\neg\}$  c) $\{\to\}$  I have looked at the answer here for (b) : How do users navigate to Meta? Duplicates in sendable DE for MobileConnect How do we escape the cycle of bad college eating? Mouseover text: And yet I never stop thinking, 'sure, these ingredients cost more than a restaurant meal, but think how many meals I'll get out of them! Especially since each one will have leftovers!'    Randall Munroe really hit the nail on the head with this comic and especially the mouseover text. There's this sort of cycle that I seem to follow when I try cooking, and I need to get my way out of it. It goes like this:   Decide I should cook something. Thumb through my recipe books, find one or two recipes I like. Write down the ingredients for the recipes in a list. Go to the grocery store, stumble around for a while and eventually find exactly those ingredients. Go back home; at some point in time soon after, cook each recipe. Whew, that was a lot of work! Spend the next week or two eating junk again, until I decide I should cook something. (repeat)   Very similar to the comic; sometimes I too let the leftovers go bad (usually not). And every time I rack up a huge bill at the supermarket and think to myself it's justified because of the leftovers or the amount of food. Not to mention, that's a lot of work for just a couple of meals, and a long list of ingredients.  I love cooking fresh meals but that's just so much overhead and so time consuming! So I always resort back to Easy Mac, Ramen, frozen dinners and frozen pizza. Yuck.  How can I escape this cycle and cut down on the overhead? How can I focus on more than 2 recipes at a time without buying giant lists of ingredients and spending tons of money? Is it really worth the trouble or should I just chock it up to being a college student and wonder about the years ahead of me where I'll have much more time to spend cooking one meal at a time? Why can some holodeck matter leave the holodeck, while other holodeck matter (notably people) cannot? I want to learn to use biper and biblatex in Lyx. This is how I would do it in LaTeX traditionally with biblatex and biber  \documentclass[12pt,a4paper,oneside,pdftex]{report} \usepackage[backend=biber]{biblatex} \addbibresource{minCuts.bib} ... \printbibliography \end{document}   The error is apparently originating from the \bibliographystyle command and biblatex package getting mixed, by . But I have no \bibliographystyle command in Preamble so they should not be mixed unless Lyx is somehow importing the \bibliographystyle.   How to resolve the biblioraphystyle error in Lyx with Biber and Biblatex?    Settings by the instructions  and Error        Document > Settings > Bibliography > Biber                  In Document > Settings > LaTeX Preamble                   Insert > List / TOC > BibTeX Bibliography...                        Package biblatex Error : '\bibliographystyle' invalid. This questions has already been asked a million times, but nothing solved my issue so far.  Situation   I'm using Blender 2.72b.  I'm not using nodes, just the plain old vanilla settings. But apparently I messed them up somehow. I'm using the Mirror modifier. I'm completely new to blender (and 3D modelling in general) and use it for a couple of days now. I deleted the cube and made this model from scratch (no import).   In texture view everything looks fine:   In material view (and after rendering) everything is black:   Attempts to solve   I have a light source. I have a camera. I have no ceiling. I have a material. I have a texture. Shadows are activated. I tried using nodes and my output was well connected - no change, still black, so I stopped using nodes again (I had no idea what I was doing anyway). I checked out many solutions posted on the web.   As my last attempt, I opened the vanilla start-up file again. Everything works fine with the cube. So I compared the material und texture settings step by step. It all seems the same. This topic is so extraordinary complex and I'm so bloody inexperienced that I'm stuck at this point. Intuitive analysis of gradient, divergence, curl FireFox 66 Quantum disabled all my extensions at midnight! How to partition harddisk for Ubuntu 18.04 I want to partition my hard disk, but I don't have any other software running(like dual booted).  And I do not know how to do it. Can you send the instructions? Clipping raster with mask in QGIS gives AAIGrid error? Should I say "She is in the park" or "She is at the park"? I am really confused.  Which preposition is correct?     She is in/at the park.   They are in/at the park.   I am in/at the park.   Should I use in or at in these sentences? Are there any holidays in Middle Earth, except Yule, Lithe, New Year in Gondor and Durin's Day? Adding console.log to every function automatically Bug in editor for numbered lists? I noticed what seemed to be a bug in the way the stackoverflow editor handles numbered lists. I did a search on meta for this and it didn't come up, but if it is a duplicate, I'll delete the question.  Consider the following code:  Here are some points:  1. One 2. Two 3. Three  Here are some more points which continue the theme:  4. Four 5. Five   When you actually type this into the text editor, this is the result:  Here are some points:   One Two Three   Here are some more points which continue the theme:   Four Five   In this case, the ordered list resets after the break. This is probably just an edge case (and not necessarily worth fixing), but I ran into the problem today and figured this would be a good place to raise the topic! My workaround was to use back-ticks around the numeral, but then you lose the hanging indent. (ie, 1. One)  Might this be worth fixing? Is it expected behavior? Or am I doing it wrong? Construct a command by putting a string into a tty I managed to do this  echo -n "command" &gt; /dev/tty1  The letters appear, and the cursor moves, but they are "ghosts" – if you hit Enter, nothing happens (they are not in stdin).  Edit:  In the middle of the below screenshot, you see why I see the use of this. (The line with a red caption, right below the line with a yellow caption.) As it is now, you are not really "editing" the note text; you are just asked to write a new text, which will replace the text of the note you are (not really) editing. Thus, I thought it could be remedied by simply pasting the old text into the tty: if the user hits enter, no modification is made. (This program is in Perl/MySQL, but I thought it would be more interesting to ask for a general solution than "how do I do this in Perl".)    Edit 2:  Here is the Perl code, that uses the C code below (works exactly as intended), as well as a new screenshot – hopefully this will clarify things beyond doubt :) Again, look at the middle of the screenshot, where the edit is made to the note text - this time around, the old text is there, for example if you just wanted to fix a typo, you won't have to retype the entire note text.  my $edit_note_text = $edit_note_data[2]; print BOLD, RED, " new text: ", RESET; system("writevt /dev/tty \"$edit_note_text\""); my $new_text = &lt;$in&gt;; $new_text = fix_input($new_text); my $set_text = "UPDATE notes SET note = \"$new_text\" WHERE id = $edit_note_id"; $db-&gt;do($set_text); Accessing site-specific Help Center from iOS app Can the Stack Exchange developers add an option to access the site-specific Help Center via the iOS app?  My use case is when I read a question that I think is off-topic, and want to double-check that the wording from the site's Help supports that position, having the Help Center available would be nice. Having trouble finding Plantera's Bulb I have been playing on hardmode for a while and have beaten the three  a couple of times. But I have yet to find a . It's getting really frustrating.  Does anyone have any methods to finding a Plantera Bulb? How to test if my server is vulnerable to the ShellShock bug? How can I ensure my Bash installation is not vulnerable to the  bug anymore after the updates?