More clearly the question is:  Prove that R is a field by showing that $\mathbb{R}$ forms an isomorphisim to $\mathbb{C}$ as rings (explicitly find an isomorphism between them and prove it is an isomorphism).  Let $$R=\left\{\begin{bmatrix}a&amp;b\\-b&amp;a\end{bmatrix}:a,b\in\mathbb{R}\right\}\subseteq M_2(\mathbb{R}).$$  So I need to show that this is a bijective homomorphisim. The definition that is given is:  Definition 4.1.7: A homomorphism $\mathbb{\phi}$:𝐺 → 𝐺′ is injective if $\mathbb{\phi}$(𝑔1) = $\mathbb{\phi}$(𝑔2) implies 𝑔1 = 𝑔2 for any 𝑔1,𝑔2 ∈ 𝐺. It is surjective if for every 𝑔′ ∈ 𝐺′, there is a 𝑔 ∈ 𝐺 such that $\mathbb{\phi}$(𝑔) = 𝑔′. It is bijective if it is both surjective and injective.   Any hints would be greatly appreciated. To be honest I am not even sure what the question is asking. If someone could break this down it would be great. How do I show all the properties of a homomorphisism from the matrix of reals to the matrix of the complex (if that is even what is being asked)? I am even more confused on showing it is injective. This is my first time encountering material like this before. It is very rough.
Suppose $K \subset L$ is a finite field extension of degree $m$. Is it true that there exists some natural $n$ such that $L$ is isomorphic to a subfield of $M^{n\times n}(K)$, the ring of $n\times n$ matrices with entries in $K$? What is the smallest dimension (as a function of $m$) for which an isomorphism exists?  The example I had in mind was the 2-dimensional extension $\mathbb{R} \subset \mathbb{C}$ with $\mathbb{C}$ being isomorphic to $\{ \begin{pmatrix}a&amp;-b\\b&amp;a\end{pmatrix} : a, b \in \mathbb{R}\}$.
Point reflection over a line I'm having trouble understanding the solution presented below (It's from a textbook). I tried to get something similar, but to no avail. Help me find the way he derived those a,b,x2 and y2 expressions.   Point p is a point which we want to reflect (It's a simple structure with two integers x and y representing coordinates on a 2d plane); (x0,y0) and (x1,y1) are coordinates of endpoints of a line.  Expected result is a new point with reflected coordinates.   Point mirror(Point p, int x0, int y0, int x1, int y1) {     double dx,dy,a,b;    long x2,y2;    Point p1; //reflected point to be returned      dx  = (double) (x1 - x0);    dy  = (double (y1 - y0);     a   = (dx * dx - dy * dy) / (dx * dx + dy*dy);    b   = 2 * dx * dy / (dx*dx + dy*dy);     x2  = Math.round(a * (p.x - x0) + b*(p.y - y0) + x0);     y2  = Math.round(b * (p.x - x0) - a*(p.y - y0) + y0);     p1 = Point((int)x2,(int)y2);      return p1;  } Why does an unhinged body rotate about its centre of mass? Cellcolor overwrites partial horizontal lines (\cline) So I need to make a colored table with multirows, together with the use of partial horizontal lines. I used \cline which does not work: the \cellcolor-command overwrites the line.  told me to change it to \hhline which will not be overwritten by \cellcolor. Instead, \hhline overwrites the color.  To make it more clear:  Without color:  \documentclass{article}  \usepackage{colortbl} \usepackage{multirow}  \begin{document}    \begin{tabular}{|c|c|}     \hline     \bfseries ColumnOne &amp; \bfseries ColumnTwo\\ \hline     First data &amp; 932\\ \hline      &amp; 239\\ \cline{2-2}  % or \hhline{|~|-|}      &amp; 137\\ \cline{2-2}  % or \hhline{|~|-|}     \multirow{-3}{*}{More data} &amp; 319\\ \hline     Last data &amp; 132\\ \hline   \end{tabular}  \end{document}     With color: Using \cline  \documentclass{article}  \usepackage{colortbl} \usepackage{multirow}  \begin{document}    \begin{tabular}{|&gt;{\columncolor{red}}c|&gt;{\columncolor{blue}}c|}     \hline     \bfseries ColumnOne &amp; \bfseries ColumnTwo\\ \hline     First data &amp; 932\\ \hline      &amp; 239\\ \cline{2-2}      &amp; 137\\ \cline{2-2}     \multirow{-3}{*}{More data} &amp; 319\\ \hline     Last data &amp; 132\\ \hline   \end{tabular}  \end{document}    Makes the lines in the right column disappear.  Using \hhline  \documentclass{article}  \usepackage{colortbl} \usepackage{multirow} \usepackage{hhline}  \begin{document}    \begin{tabular}{|&gt;{\columncolor{red}}c|&gt;{\columncolor{blue}}c|}     \hline     \bfseries ColumnOne &amp; \bfseries ColumnTwo\\ \hline     First data &amp; 932\\ \hline      &amp; 239\\ \hhline{|~|-|}      &amp; 137\\ \hhline{|~|-|}     \multirow{-3}{*}{More data} &amp; 319\\ \hline     Last data &amp; 132\\ \hline   \end{tabular}  \end{document}    Creates white lines on the left column.  Is there any possible way on how to resolve this? I recently received an invitation to a conference and I'm trying to determine how reputable it is. Is there a good way to go about this? How can I see my progress for different badges? I am trying to obtain more badges on Stack Overflow. For example, to gain Archaeologist I need to edit 100 inactive posts but I am not keeping track of the number of posts I've edited. For Deputy I need to raise 80 helpful flags, etc. How can I view all my statistics about the badges, such as how many more edits I need to do, etc. I'm not doing it for badges, but I can spend a bit more effort if I'm close to getting a badge. How to quit Skype on Android? Prove that for every $a\in A$ $\exists$ $q\in M,s\in A$ s.t. $a=sba+qa$ where $M$ is maximal ideal in $A$. Let A be a commutative ring with unity ring, M be a maximal ideal and $b\notin M$ . Show that there exists $q \in M, s\in A$ such that for every $a \in A$ we have $a=sba+qa$. Show that this is true only when $M$ is a maximal ideal.   Suppose if we take $A=2\mathbb Z$ and $M=4\mathbb Z$, $a=6.$ Then by the given statement $\exists$ $s\in A$, $q\in M$ s.t. $6=6(2s+q)$ $\implies$$s=\frac{1-q}{2}\implies q \text { is odd}$. But $q \in 4\mathbb Z.$  Please someone give some hints. how can solve this.  Thank you.. How can I achieve more clarity in my photos of the moon? I took this photo of the blood moon this morning and I am finding many of my photos, especially landscape photos, have a significant lack of clarity. I'm not looking for a photo critique because I know this particular photo is crap. I'm looking to identify my source of failure to nail a significantly clear photo.     Setup  Canon EOS 70D with a 100-300 telephoto lens (I'm sorry I don't know the exact model. It's from Canon but not one of their professional quality lenses.)   Aperture: f/5.6 Shutter: 1 sec. ISO: 1000 Focal Length: 300mm Metering mode: Spot White Balance: I manually set the WB to 5500 kelvin, it looked fine in live view. Focus Mode: Manual Drive: 2 sec timer delay   Mounted on a decently stable tripod  Process  I turn on live view, and square up the target in the display. I then set my focal length, focus in using live view, adjust the focus as sharp as possible and take the pic. I've noticed that in this case, the picture was incredibly grainy, which doesn't translate to the final photo, but wonder if it has anything to do with it.  This particular shoot I found it difficult to balance out the variables with a dark sky and a moving moon that starts blurring after 1-2 seconds of exposure, which is why I had the aperture wide open and the ISO cranked up.  I can't tell where I'm failing:  Equipment:  The 70d takes great photos but the telephoto is not great when it's totally zoomed in. My wallet doesn't hold enough cash for a motorized mount to track objects in the sky.  Circumstance: I've seen great photos of the moon with the use of a camera and a tripod, so I know this is possible. Although, I keep seeing pictures of this big gorgeous moon over the landscape, and when I use my wide-angle, the moon shows up as a dot. What the hell are people doing to get the moon so big?  Beginner skill level: I know this is the biggest culprit, but being the beginner, I'm unable to know where to adjust. options for appearance of links in hyperref If I use  \usepackage[colorlinks]{hyperref}   I get colored links for \cite (not good for printing or even viewing) but table of content looks fine.  If I use  \usepackage[colorlinks=false]{hyperref}   I get \cite  links in black with a colored frame (this looks very good) but the table of content looks strange with extra long boxes.  What are my other options? Can you have two styles in different parts of a document?   How do you make the page numbers in TOC to be the links instead of the section headers? How to do a bias-variance analysis on a machine learning modelling process How can I extract text from images?  I am not talking about scanned files, but garden variety images, such as when you take a high-def picture of a blackboard at class, and it is nicely handwritten; or when you photograph a page from a recipe book and want the recipe in text format.  Any free and open software for that?  I tried tesseract, and the results were awful. That's a lot of ___ vs. those are a lot of ___ That's a lot of cars!   Those are a lot of cars!      The context is when a someone sees many objects (in this example, cars) and makes the exclamation.   According to , lot can handle either singular or plural verbs.  I agree with the author who states that that's sounds better, but the post doesn't seem definitive either way.     Are both grammatically correct and do they have the same meaning? Is the W statistic output by wilcox.test() in R the same as the U statistic? I've recently been reading about the Mann-Whitney U test. It turns out that to carry out this test in R you actually need to run a Wilcoxon test!  My question: is the W statistic of wilcox.test in R identical to the U statistic? Where is attribute table in ArcGIS Desktop when not visible? I've just updated to ArcGIS 10.1 for Desktop and when I go to open an attribute table, I can not see it anywhere.   I'm wondering if it is off my screen somewhere or if its just super tiny that I can't see it.   I'm using dual monitors and it's not popping up on either.  Is there a way I can find my attribute table? How to round a number to n decimal places in Java I searched wide and far and couldn't come up with a title for a movie I watched somewhere around the mid-90ies up to early 00's as a rerun in late-night television (in Germany, specifically on Vox; but I'm sure it was an American movie to begin with).  Basically, what I remember from the story revolves around a crew in a space station or maybe space ship of some sort. Some crew members murder other crew members and there is some kind of door they are not supposed/unable to go through. Late in the movie they do go through the door and I'm pretty sure one of the first views is a skeleton of a dead security guard. It turns out the whole thing was some kind of simulation gone wrong and they find themselves to be in a (nuclear?) wasteland. And that's pretty much the revelation/wrap-up of the movie, as I recall it.  I would be delighted if somebody had any idea what movie this could be :) Thanks!  EDIT: The whole movie was in color and I think it was feature-length, not just 45 mins. Every word is marked as misspelled in TeXstudio Why is that most of the words, almost 90 percent, are marked as misspelled words when there is no error? And the spellchecking suggestions don't even look like English.  I am using TeXstudio as my editor. Converting bike from rim brakes to disc brakes I just bought a used  and it comes with regular rim brakes. I'm perfectly happy with them so far, but have been wondering if disc brakes would come handy when it rains.  What is involved in converting a bike from rim brakes to disc brakes (mechanical or hydraulic)?   Is this something that is feasible, or am I better off buying another bike that already has disc brakes? All of a sudden, Safari on my Mac has started misbehaving. I can't select the text in the address bar, and so I can't go to any new websites. When it redraws, it goes all weird:    I've tried deleting the cache, clearing all cookies, disabling all plugins and rebooting. These things didn't help. I've had to switch to Chrome for the time being. Can anyone suggest some other possible remedies? Shift numbers into a different range I was wondering how can I shift my data that fall between a range lets say [0, 125] to another range like [-128, 128].  Thanks for any help