Let $V$ be a finite-dimensional vector space over field $k$ and $R = \text{End}_k V$. How do I see that any left ideal of $R$ takes on the form $Rr$ for some suitable element $r \in R$?   Since $R$ is a semisimple ring, every left ideal with split like this: $R = L \oplus L'$. In this decomposition, $1 = e + e'$ where $e \in L$ and $e' \in L'$.  Why does $e^2 = e$ and $(e')^2 = e'$?  And why does $L = Re$?
As the question title suggests, how do I see that every left ideal of a square matrix ring over a field is principal?
How can I visualize or simulate the effect of different focal lengths? I have seen many camera review sites illustrate the effect of using different focal lengths on the same photograph using frames. There are also similar illustrations to compare different sensors (). I find this kind of visualization to be very useful in comparing the actual effect of the longer focal length\different sensor size. Of course the ability to do a wider focal length visualization would also be equally useful, but that's not possible.  Is there any ready-made software or plugin that enables this? If not, is there any simple technique to visualize the effect of longer focal lengths on an image (one would of course have to use the zoom information in the image EXIF data to accurately represent this)? All I can think of is to use some trigonometry to do the necessary calculations for the frame sizes for cropping. With no good reason I had a look at my computer's storage going on the apple and About this Mac. Here is the display:    The pink bar doesn't make much sense. It obviously does not represent the 80.67GB of free storage. Two sections correspond to pink; "Photos" and "Backups". I obviously don't have 140.76TB of backups. And probably not on this computer that has only 320GB of storage.  What is going on here? Blade of the Ruined King Vs. The Bloodthirster Effects:  +25 attack damage +15% life steal +40% attack speed Passive  Unique: Your attacks deal 5% of the target's current health in physical damage (60 max vs minions and monsters). Active   Unique: Drains target champion, dealing 15% of the champion's maximum Health in physical damage and healing you by the same amount. Additionally you steal 30% of their movement speed for 4 seconds - 60 second cooldown. (500 range)  Cost: 3200g VS.   Effects:  +70 attack damage +12% life steal Passive: Gain +1 attack damage and +0.2% life steal per kill (maximum +30 attack damage and +6% life steal). Half of the bonuses are lost upon death.  Cost: 3200g What is better for what situations, and when should you build BT instead of BotRK and Vice Versa. Which provides more lifesteal, the only variable being the stats of the weapon? I find myself constantly questioning my decision on which weapon to get and would really love some in depth answers. Would be great if any examples given were given from the ADC side of the house. How align the qedsymbol after of a big formula in display mode? Description of the problem My problem is this. For the proof of a math theorem i use the proof environment of the package amsthm. It adds a \qedsymbol in the last line of the proof, but if the proof ends with a displayed formula the qedsymbol appears in the next line leaving an unwanted blank space. The documentation of  teaches to fix this problem placing the command \qedhere after of the displayed formula but inside of the appropriate environment. This works well if the formula is of one line like \[ax^2+bx+c=0.\qedhere \]  but if the proof ends in a matrix, or cases with a &quot;great height&quot; the qedsymbol is fixed in the baseline at right and it this looks higher than expected (for my taste). The result is showed in this MWE \documentclass{article} \usepackage[width=7cm]{geometry} % Only for reducing space in this MWE  \usepackage{amsmath} % For `bmatrix` \usepackage{amsthm} % For `proof` environment \begin{document} \begin{proof} And this proof ends with \[ \begin{bmatrix}     1 &amp; 0 \\     0 &amp; 1 \\     0 &amp; 0 \end{bmatrix}.\qedhere \] \end{proof} \noindent Text \end{document}  that produces  What I want I'd like something like  with the qedsymbol aligned with the bottom of the matrix. Is this possible? How's it fair for a moderator to delete my account and dissociate my questions and answers? Edit I found the email that proves a moderator deleted my account. And nobody appears to believe me before? Even if you suspend me, why dissociate my questions and answers? How can I manage or track them, or accept answers?  Original  Moderator messaged me at Personal Finance S.E. How am I supposed to see if when they deleted my account? When I click on &quot;moderator message&quot;, I get 404 not found page.  Why is this being down-voted?   Many people don't believe me that a moderator deleted my old account.  Yesterday when I logged on, I saw &quot;Join this community&quot;, but I never deleted my Money account. I then clicked &quot;Join this community&quot; to see what happened.  Screen shot beneath undeniably proves this wasn't my old account when I asked at least 5 questions. It's unquestionably false that &quot;you have not asked any questions.&quot;  Please reinstate my old account? I want to keep track of my posts.   I can't post on Money Meta. Please don't betray private details.    and: I have two different list. In the first list are some people picker columns.  Is it possible to create a lookup column for list 2 to add the user from the first list? I know it isn't supported in Sharepoint, but i hope someone have an idea to solve my problem. From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude.  I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation.  See also: Urgent Sydney transit visa question Can we have a perfect field of characteristic $p$ with a non-archimedean valuation? Why aren't the braces showing "Can’t help but" vs. "can help but" How do I delete totem's history? How do I delete totem's history displayed in it's main menu in Natty? For example if I don't want anyone to know I'm watching... ehem Simpsons a lot?  I've tried  but this doesnt work for Totem. How to extract the last inserted row in SQL Server? When an insert statement is executed one or more rows is inserted into the table, is there any way to extract the last inserted row in SQL Server? I am aware    but I ask this regarding SE-2.0 sites, as the example I will take gaming:  Both answers on the question  make perfect sense to me, i.e. on the one hand this might overuse the tagging system but on the other hand it would help a lot for filtering.  Therefore I suggest a system which, in this case, would internally store the platforms to which a tag is relevant and   if a question is tagged with one of these platforms, nothing special will happen and the question is to be considered platform specific (e.g. a bug which only appears on the PS3 version) if the question is not tagged with a platform, the only result will be that the tag filter mechanism works a bit different compared to what would happen if the question was (invisibly) tagged with all platform tags:   If the invisible tagging is definite on interesting or ignore (i.e. there is not at least one ignored and one interesting tag) the usual mechanism works. if the tagging is ambiguous (i.e. contains both interesting and ignored tags), no filtering happens at all    The actually attributed tags must of course override this behaviour. That way questions about a game one likes but which is available for a platform one doesn't care about will still be highlighted unless specifically tagged with that platform, but no additional tags have to be used to filter a game which is only available on ignored platforms.  This can of course easily be extended to other sites, e.g. the  may be interesting both general and for c++ but totally uninteresting if the question is only about a JavaScript implementation.  I am only unsure whether search should include those hidden tags, too. What is "cloning a hard drive" and how does one go about doing it? What does cloning a hard drive mean? Is it just simply copying over all the "CONTENT" of the hard drive EXCLUDING the operating system, drivers, all the installed programs? or is it copying the ENTIRE hard drive BIT for BIT? Is the cloned hard drive bootable?  I have an external USB hard drive and trying to make a exact copy of my computer system by cloning, so that I can connect the cloned hard drive to another machine and boot into it, or install the cloned disk on another computer system and boot into it. Is this possible? Or am I going about this totally wrong?  What is ghosting a hard drive? I'm more of a math guy than a physics guy so bear with me....  In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal dimension of around $1.2$. You can also define a concept called fractal extent (which is the experimental version of hausdorff measure). This extent is a measure of how large a fractal is. It's even been used to more accurately model the distribution of coastline bird nesting.   Even though fractals work as an interpretation of geometrical units, I don't think they would describe things such as $kg^{1/2}$ very well. In addition, I've heard about white Gaussian noise that has square root time units.   What is the physical meaning of fractional units such as the above? (Since it obviously has little geometric meaning) How to install Windows 7 after Ubuntu and dual boot? Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$ Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$.   I was thinking among the lines of:  Since $a$ and $b$ are coprime numbers, $\gcd(a,b)=1$. Then there exist integers $x$ and $y$ such that, $ax+by=1$.   Then, $a=(1-by)/x$, $b=(1-ax)/y$  So if I write $2^a+1$ as:  $2^{(1-by)/x}+1$  Then can I say that the above expression is equivalent to 0 (mod 3)? Is $v=\ (r_i)^{-1}\cdot z $, a uniformly random value of a field? We consider a finite field $\mathbb{F}_q$ where $q=2p+1$ and $q$ and $p$ are prime numbers.  Let $r_i$ be a value picked uniformly at random from the field such that $r_i&gt;\frac{q}{2}$. Let $z$ be a fixed field element.   Question: Is $v=(\ (r_i)^{-1}\cdot z) \bmod q$, a uniformly random value of the field? I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the following limit: $$\lim_{x\to\infty}{\frac{F_n}{F_{n+1}}}=?$$ Where $F_n$ is the nth Fibonacci number?