Finding the right ideals of $M_{n\times n}(\mathbb{F})$ (matrices over a field $\mathbb{F}$)
What are the left and right ideals of matrix ring? How about the two sided ideals?
Where To start Learning With Salesforce Development If $A$ is a non-empty set, then $A \notin A$ Exercise 3.2.2 Use the axiom of regularity and the singleton set axiom to show that if $A$ is a set then $A \notin A$.  Axiom of Regularity: If $A$ is a non-empty set, then there is at least one element x of A which is either not a set, or is disjoint from A.  Singleton Set Axiom: If $a$ is an object, then there exists a set $\{a\}$ whose only element is $a$, we refer to $\{a\}$ as the singleton set whose element is $a$ Furthermore, if $a$ and $b$ are objects, then there exists a set $\{a,b\}$ whose only elements are $a$ and $b$    I think I can give the proof when A is the non-empty set which contains finite elements. However, can anyone give any hint for the general case?  Suppose $A \in A$   Then eixsts $x$ such that either $x$ is not a set or $x \cap A$=$\emptyset$  Then it is clear thar $x \ne A$ (if $x = A$, then $x \cap A \ne \emptyset$)  We observe that as an element, $A$ $\ne$ $x$, $A$ $\in$ $A$  Then $A$ $\in$ $A-x$   Define the new set $A-x$ as $A_1$  then exists $x_1$ such that $A \in A_1-x_1$  After finite steps  we find that $A \in \emptyset $  which leads to contradiction. Problem 1.1.5 from Hatcher's I want your advice on my solution of this problem.  problem  Show that for a space $X$, the following three are equivalent:  (a) Every map $S^1 \rightarrow X$ is homotopic to a constant map, with image a point.  (b) Every map $S^1 \rightarrow X$ extends to a map $D^2 \rightarrow X$.  (c) $\pi_1(X,x_0)=0$ $\forall x_0 \in X$.  my solution  $(c) \rightarrow (a)$: Take a map $f:S^1\rightarrow X$, and the constant loop at $x_0$ (which from (c) we know there's only this loop), and take the linear homotopy connecting $f$ and $x_0$.  $(b)\rightarrow (c)$: I am not sure if this is right, but I want somehow to induce some isomorphism from $\pi_1(D^2,y_0)$ to $\pi_1(X,x_0)$ in which case we'll get the required result. So I argued that we take as a map the extension of a map $S^1\rightarrow X$ to a map $D^2\rightarrow X$, but I don't see how to argue that the last map is homeomorphism, any advice?  $(a) \rightarrow (b)$: I am given $f:S^1\rightarrow X$, and it's homotopic to some point $x_0$, so I thought to extend $f$ on the whole ball, by defining the map: $g(x)= f(x)$ if $|x|=1$ and $g(x)=x_0$ if $|x|&lt;1$.  Is any of what I typed right, what do I need to amend?  Thanks. I have two cameras, a 6mpix Panasonic FZ8, and 14mpix Canon A2200.  When I compare shots from two cameras, one of differences I notice is dpi number that camera reports in JPG info:   FZ8 reports dpi=72 A2200 reports dpi=180   What physical meaning can these numbers have? I am at loss for guesses.    I am fairly familiar with notion of dpi in scanning and printing. I can calculate density of pixels on the sensor of the camera. But then, the linear density of pixels on the sensor will be hundreds times larger than number above. So what, if anything, does it mean? I am having trouble finding out why the white space between my paragraphs is variable. Sorry, I am not allowed to post images but sometimes there is no white space (like it should be), just an indent to denote that a new paragraph has started. While sometimes the paragraphs start with an indent AND there is (sometimes substantial and thus variable) white space between the last sentence of the previous paragraph and the first sentence of the new paragraph.  In my tex file, there is no difference between what I do in the first and second examples of my text: just two hard returns to denote the start of a new paragraph. I was wondering whether it could be due to placement of figures and footnotes but I have no idea how to change it. I don't mind a little variable white space between paragraphs but the above example is too extreme for my taste. Thanks for your advice! Since drush 9 (drupal 8.4 requirement), alias @sites does not seem to be implemented anymore (or yet ?).  $ drush @sites cr The alias @sites could not be found.   Is this a way to get the same behavior that does not consist of typing my 300 alias in a config file ? For some special purpose, I want to prevent non-root users of the Linux Server from changing or renaming the filenames. However, they can modify and write to the contents of the file. How to do this from command line. Is there something like python's interactive REPL mode, but for Java? Is there something like python's interactive REPL mode, but for Java? So that I can, for example, type InetAddress.getAllByName( localHostName ) in a window, and immediately get results, without all this public static void nightmare() thing? I just finished backing up my system and when I try to upgrade to Utopic Unicorn, I get the error "Cannot Calculate the upgrades." I ran the upgrade by sudo apt-get dist-upgrade, sudo do-release-upgrade after the GUI method failed, they all gave the same error.  The feedback gives some recommendations:  This can be caused by:  * Upgrading to a pre-release version of Ubuntu  * Running the current pre-release version of Ubuntu  * Unofficial software packages not provided by Ubuntu    I am not upgrading to a pre-release version nor coming from one, I'm going from Ubuntu 14.04 LTS to 14.10. I have no broken packages and I have unchecked all my non-official software sources in the 'Software &amp; Updates' manager.  My guess left is the Unofficial software packages not provided by Ubuntu. If anyone could give any help as to how to check for unofficial packages using Synaptic or another, my appreciation.  Are there any other factors that may be causing this error and how would I go about checking/fixing those?  Thanks.  Edit: I have no broken or held packages. Everything is up-to-date, and restarting everything or anything has no effect. How do I type Arabic text in Photoshop? I want to type Arabic text in Photoshop, but the text appears disjointed and the letters do not connect at all. For example محمد appears as م ح م د.   I'm using Photoshop CS6 on OSX Mountain Lion. Would a Bayesian admit that there is one fixed parameter value? In Bayesian data analysis, parameters are treated as random variables. This stems from the Bayesian subjective conceptualization of probability. But do Bayesians theoretically acknowledge that there is one true fixed parameter value out in the 'real world?'  It seems like the obvious answer is 'yes', because then trying to estimate the parameter would almost be nonsensical. An academic citation for this answer would be greatly appreciated. Cannot change stroke width in Inkscape I am trying to change the stroke width of an icon in Inkscape but, unlike other times I have done this, this time the stroke width appears to be 0 px and I am not allowed to make it thinner:    The stroke width is actually quite large already, is there another setting I should edit to make it thinner?  You can download the file I am working on .  BTW: sorry for the potato quality of the screenshot, this is what you get using Inkscape on a retina screen :/ Edit: The previous title didn't really ask the same thing as the question (sorry about that), so I've changed it. To clarify, I understand that the action isn't always a minimum. My questions are in the points 1. and 2. below.  I understand that &quot;principle of least action&quot; is somewhat of a misnomer, since we find that to determine the path taken by a system, we need only impose the condition that the action be stationary, i.e., that $\delta S$ should vanish to first order for small variations of the path, and this leads to the Euler-Lagrange equations. In The Classical Theory of Fields, Landau &amp; Lifshitz discuss the relativistic action for a free particle:  So for a free particle the action must have the form $$S = -\alpha \int_a^b ds$$ (...) It is easy to see that $\alpha$ must be a positive quantity for all particles. In fact, as we saw [earlier], $\int_a^b ds$ has its maximum value along a straight world line; by integrating along a curved world line we can make the integral arbitrarily small. Thus the integral $\int_a^b ds$ with the positive sign cannot have a minimum; with the opposite sign it clearly has a minimum, along the straight world line.  There is also a footnote, addressed a couple of paragraphs earlier, but which is relevant:  Strictly speaking, the principle of least action asserts that the integral $S$ must be a minimum only for infinitesimal lengths of the path of integration. For paths of arbitrary length we can say only that $S$ must be an extremum, not necessarily a minimum.  I have two questions regarding this:  How is the condition &quot;the action must be a minimum for infinitesimal displacements&quot; formulated? I've never heard of that outside of Landau &amp; Lifshitz's books, and in Mechanics they mention it as well but doesn't go into detail. Is this discussed a bit more somewhere?  If for the whole path the action only needs to be stationary, how can we make the argument for the negative sign? If the action had to be a minimum then it would make sense, but surely the fact that $\delta S$ = 0 isn't affected by an overall sign? Background: I'm making a retro style game for both fun, and an unofficial contest.   I have bought a song, song X. However, I then found notation for song X, and all the instruments. So basically, it's a program recreating song X in a sense.  I took the notation, and used another program to make it sound like a chiptune does (just for the fun of it, and it sounds good :P). However, I would like to use this in my game. It's non-commercial, but is this allowed? I would've said yes, because I'm not actually using the original, copyrighted song.  I think the main point is can I:   Use the music with no reference to the author Use the music with a reference, since it's non-profit. Not use the music at all since the melody and rhythm is copyright. How a module can override theme suggestion? Basically it was  but I'll reword it, as I don't want to define a new theme (reason was already given).  What I want to do: in my module I have a file node--attach.tpl.php which is a modifed version of node.tpl.php. Now I want that, if view_mode = 'attach' then this file is used. Also, theme should be allowed to override it with its own node--attach.tpl.php in theme folder. Correct way to use include and includeonly when writing a large document like a thesis I am writing a large document with several chapters and I would like to work on each chapter separately. In my main.tex I include the chapters either like   \include{chap1.tex}  \include{chap2.tex}  \include{chap3.tex}   or   \input{chap1.tex}  \input{chap2.tex}  \input{chap3.tex}   I think \include is preferable if I want things to be referenced and cited correctly? but doing it the \include way latex main.tex I don't get any dvi output whereas with \input things compile fine to dvi. What's going wrong here?  I would then like to use \includeonly (or \excludeonly ) to work on individual sections of the document whilst maintaining the referencing etc.  A mwe can be handy:  \documentclass[a4paper,10pt]{article} \includeonly{chap1}  \begin{document}   \include{chap1}   \include{chap2}   \include{chap3} \end{document} Combining all the address fields into one? How do I use formatting in comments? I see some people write comments that are for example - in italics.  I do not see any way of doing that while writing my comments. Why is it so? How to correctly post a Q&A style question (i.e.self answer) without it failing minimal understanding As I understand it Q&amp;A style questions are  but a self answered question should be judged independently of the answer. Often after I've solved a problem I would like to share the question and answer on stack overflow (and have on a couple of occasions) but how to show &quot;minimal understanding of the problem&quot; is always challenging. For example I have been looking into the following question  How can I plot a straight line in the JMonkey Engine library I'm trying to plot straight lines between vertices that I specify using the JMonkey Engine 3D graphics library. So for example if I was to try to plot between: (2,0,0) (-1,0,1) (0,1,1) (1,1,1) (1,4,0) Then I would get:   This is something I do (and forget how to do) relatively often and the answer is harder to find than you'd imagine ( provides everything needed to figure it out but its spread out over several posts and isn't a complete solution on its own). Therefore I want to provide a Q&amp;A for it but at present this question feels like it would fail the minimal understanding test  Questions must demonstrate a minimal understanding of the problem being solved. Tell us what you've tried to do, why it didn't work, and how it should work  Should I make up some imaginary problem? (Probably the final source code messed up a bit) Or mention in the question that I’m self answering to assuage some of the normal &quot;This is not a do your work for you site&quot; comments. How do I "Show total up and down vote counts" since I'm at 1K rep?