Is it correct that set ${\mathbb C}$ is isomorphic to the set of following 2x2 matrices: $$\left( \begin{array}{cc} a &amp;-b\\ b &amp;a  \end{array}\right) $$ $a \in {\mathbb R}$ and $b \in {\mathbb R} $?   In other words: are these two sets identical?
Question. Let $M = \left\{ \begin{bmatrix} a &amp; b \\ -b &amp; a \end{bmatrix} \,\middle\vert\,  a, b \in \mathbb R \right\}$.  Show that $(M,+)$ and $(\mathbb C,+)$ are isomorphic.  Show that $(M^{*},*)$ and $(\mathbb C^{*},*)$ are isomorphic.    (Here, $M^{*}$ is defined as excluding the zero-element) I understand that there exists an isomorphism $\phi:M \rightarrow\mathbb C$ defined by $\begin{bmatrix} \alpha &amp; \beta \\ -\beta &amp; \alpha \end{bmatrix}\mapsto \alpha+i\beta$, ie. I can show that $\phi(M_{1}+M_{2}) = \phi(M_{1})+\phi(M_{2})$ and $\phi(M_{1}*M_{2}) = \phi(M_{1})*\phi(M_{2})$. However, I'm unsure how to go about the two subquestions seperately.
As a native English speaker, which two further languages would give me the most travel utility worldwide? This follows on from a travel conversation a group of us had in a hostel in London.  Assuming I am travelling randomly between countries and continents, which two languages in addition to English would provide the most usefulness in any given country?  I suspect Spanish (North America, South America) and French (Europe, Africa) but with the base understanding between French and Spanish maybe the time would be best spent learning another language?  This question is a real puzzler since the following constraints apply   Length of time required to be "functional" in the language (cost-time trade off) Chance of reaching proficiency within any meaningful timeframe (Mandarin) Regional dialects destroying proficiency (Arabic) Easier Secondary languages underpinning the society (Arabic versus French argument) Similarity of Language families (Romance) leading to wasted effort (Italian v French) Utility of the Language (German versus English when English is understood) Enjoyment is a minor consideration   Happy to consider any other thoughts but it is a really interesting topic for consideration if you are linguistically minded. Ways to go from A to B(proof) Is it possible to assign a base class object to a derived class reference with an explicit typecast in C#?.  I have tried it and it creates a run-time error. What is syntax highlighting and how does it work? I noticed that sometimes my code gets highlighted in different colors when rendered.   What is syntax highlighting? How does it work? Why isn't my code being highlighted correctly? How do I report a bug or request a new language? How do I use syntax highlighting? What languages are currently available on Stack Exchange? Automating the appearance of email addresses as images Publishing pdf manuscripts on the web with emails embedded in the file results in spam.  More generally, I am looking for a way to automate the conversion of a string to an image.  It's quite easy to solve this old problem using  (more ). The trouble is that you need a distinct file and typesetting run, and you end up with unnecessary image files that you cannot delete carelessly lest you also delete non-generated images.  Can you think of a way to do this on the fly? It's understood that people will have to type the email addresses if they need to message you, but that's the point.  More briefly/concretely...  With what can one replace #1  \documentclass{article} \newcommand\showasimage[1]{#1} \begin{document} \showasimage{joe@foo.bar} \end{document}   such that #1 appears as an image?  Edit  It would be really nice if the pdf file remains vectorial. How does Salesforce SSO call just-in-time handler methods C# List Sort by x then y Similar to , we want to sort by one element, then another.  we want to achieve the functional equivalent of   SELECT * from Table ORDER BY x, y     We have a class that contains a number of sorting functions, and we have no issues sorting by one element. For example:    public class MyClass {     public int x;     public int y; }    List&lt;MyClass&gt; MyList;  public void SortList() {     MyList.Sort( MySortingFunction ); }   And we have the following in the list:    Unsorted     Sorted(x)     Desired ---------    ---------    --------- ID   x  y    ID   x  y    ID   x  y [0]  0  1    [2]  0  2    [0]  0  1 [1]  1  1    [0]  0  1    [2]  0  2 [2]  0  2    [1]  1  1    [1]  1  1 [3]  1  2    [3]  1  2    [3]  1  2   Stable sort would be preferable, but not required. Solution that works for .Net 2.0 is welcome. I've been looking for this feature in Blender, either upon the time of creation using Edge Cut and Slide or afterwards, in just editing the mesh. I'd like to know if it's possible to evenly slide two edges away from one another in Blender?  Here's a picture of what I'd like to be able to do:    Thanks upfront!! Rules when using a Joker I had on my rack among other tiles, a black 9, 1 x blue 10, and a black 12.  On the table was a set of 3 x 10s including a Joker(no blue 10), and other sets of tiles including an available black 11.  Can I replace the joker with my blue 10 and then set out :  my black 9, the joker(representing a black 10), a black 11 from the table, and my black 12? Why is human vision restricted to 400-700 nm? Across the electromagnetic spectrum, 400-700 nm is a narrow spectrum of frequencies and focused in the region of short wavelengths. For example, radio waves cover a large range of frequencies unexploited by the visual system. So what biological reason is there that evolved us to use such a small frequency bandwidth for vision? I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it. Basically I want  \mint{python}|'banana'|   to produce straight single quotes, rather than the curly ones it normally produces (in pdflatex). Placing items before \documentclass When looking in books or at sample code, I always see \documentclass appearing first. Is that absolutely necessary?  I have created some test code which seems to compile fine:  \def\SetClass{article} \documentclass{\SetClass} \begin{document}     This is some text. \end{document}   Is there any problem with placing other items, such as \input, \usepackage, or such details before the \documentclass? I am using \if condition in a custom command in my LaTeX document. I would like to know where I can find some good explanation about \if command. I followed a  but it's just an example; there are not any explanation. I have an iPhone 6 128 GB which is not jailbroken. The current version of iOS on the phone is 8.1.3 and yesterday I wanted to update to iOS 8.2.  The download went through without any problem but while installing, it gave the error:     Unable to Install Update   An error occurred installing iOS 8.2   Tried installing after switching Off and ON and also tried Hard Reset (Holding Home button and On/off switch).  Same error displays every time.  Also the Touch ID has stopped working and I find that the Settings - Touch ID &amp; Passcode - iPhone Unlock is OFF.  When I try to switch it ON, the system displays the error     Failed   Unable to complete Touch ID setup. Please go back and try again.   Is anyone facing this problem?  I live in India. Is it possible to flash a smartphone like the Samsung Galaxy SIII or an LG Optimus 4X with a vanilla Android ROM like the one that has been flashed on Galaxy Nexus or Motorola phones (I think)?  I am asking because manufacturers like Samsung, LG, HTC tend to make ROMs for their smartphones and tablets with preinstalled launchers like Touchwiz or HTC Sense. They explain their decision by saying that they want to make their devices to look unique against their rival devices. The problem is that Android updates for their phones and tablets get delayed and there are incidents in which some devices - which are more than capable of running the upcoming version of Android - fall way behind or never get updated at all.   So, what I am basically asking here is if there is a way to install vanilla Android, excluding CyanogenMod or any other modified vanilla ROMs, on an Android based smartphone? The equivalent electric field of a magnetic field Focus in night photography Google street view URL Hi I have the address of a property and my application can launch a browser to go to . If a good match is found it will take it directly there. Is there any thing I can append to the url to make it switch to streetview without having the exact coordinates? I dont't want to code any javascript or flash. How do I delete a folder that's in use? Sometimes when I'm working in Windows I'll get this prompt when I try to delete a directory:    Folder In Use The action can't be completed because the folder is open in another program Close the folder and try again.   However, the folder isn't "in use" at all, and I don't have it open. So how can I delete it short of rebooting?