I want to prove that for matrices $A,B \in M_n (\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$ if $AB = I$ then $BA = I$.   My proof is really short so I'm not sure it's right:  If $AB = I$ then $(BA)B = B$ and therefore $BA=I$?
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.    I do not understand anything more than the following.   Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix.   If the entries of the matrix are not from a mathematical structure which supports commutativity, what can we say about this problem?  P.S.: Please avoid using the transpose and/or inverse of a matrix.
How can I get a specific line from a file? How do I drag and drop files into an application? Transcendental numbers and irrationality measure I've got a project where I'm looking to tweak the standard lead conversion process.   Triggers and lead conversion has always been funky in past projects so I was hoping to start the design off by reviewing the order of execution, but after a long google search I'm coming up empty handed.   Anyone out there know the specific trigger order for lead conversion?    Edit: I'm not asking about trigger execution order for any object, but all the objects created in the lead conversion process.  I'd think it would be something like this:        Lead Before Update Triggers (isConverted = false, convertedAccountId = null, convertContactId = null, convertedOpportunityId = null)         Account Before Insert Triggers (not fired if merged with existing account)   Account After Insert Triggers (not fired if merged with existing account)   Contact Before Insert Triggers (not fired if merged with existing contact)   Contact After Insert Triggers (not fired if merged with existing contact)   Opportunity Before Insert Triggers (not fired if no opportunity) (OpportunityContactRoles null)   Opportunity After Insert Triggers (not fired if no opportunity) (OpportunityContactRoles = convertedContactId)   Lead After Update Triggers (isConverted = true, convertedAccountId = XXX, convertedContactId = XXX, convertedOpportunityId = XXX) What is the easiest way to superscript text outside of math mode?  For example, let's say I want to write the $n^{th}$ element, but without the math mode's automatic italicization of the th.  And what if I still want the n to be in math mode, but the th outside? I am trying to use squared loss to do binary classification on a toy data set. I am using mtcars data set, use mile per gallon and weight to predict transmission type. The plot below shows the two types of transmission type data in different colors, and decision boundary generated by different loss function. The squared loss is $\sum_i (y_i-p_i)^2$ where $y_i$ is the ground truth label (0 or 1) and $p_i$ is the predicted probability $p_i=\text{Logit}^{-1}(\beta^Tx_i)$. In other words, I am replace logistic loss with squared loss in classification setting, other parts are the same. For a toy example with mtcars data, in many cases, I got a model &quot;similar&quot; to logistic regression (see following figure, with random seed 0).  But in somethings (if we do set.seed(1)), squared loss seems not working well.  What is happening here? The optimization does not converge? Logistic loss is easier to optimize comparing to squared loss? Any help would be appreciated.  Code d=mtcars[,c(&quot;am&quot;,&quot;mpg&quot;,&quot;wt&quot;)] plot(d$mpg,d$wt,col=factor(d$am)) lg_fit=glm(am~.,d, family = binomial()) abline(-lg_fit$coefficients[1]/lg_fit$coefficients[3],        -lg_fit$coefficients[2]/lg_fit$coefficients[3]) grid()  # sq loss lossSqOnBinary&lt;-function(x,y,w){   p=plogis(x %*% w)   return(sum((y-p)^2)) }  # this random seed is important for reproducing the problem set.seed(0)  x0=runif(3) x=as.matrix(cbind(1,d[,2:3])) y=d$am opt=optim(x0, lossSqOnBinary, method=&quot;BFGS&quot;, x=x,y=y)  abline(-opt$par[1]/opt$par[3],        -opt$par[2]/opt$par[3], lty=2) legend(25,5,c(&quot;logisitc loss&quot;,&quot;squared loss&quot;), lty=c(1,2)) Proving $\arg(zw)=\arg(z)+\arg(w)$ Prove by induction $\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{n}\ge\frac{2}{3}n\sqrt{n}$ for all positive integers Assumption: $\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{k}\ge\frac{2}{3}k\sqrt{k}$   Prove true for $n=k+1$ $$\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{k}+\sqrt{k+1}\ge\frac{2}{3}(k+1)\sqrt{k+1}$$  I'm upto :$$\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{k}+\sqrt{k+1}\ge\frac{2}{3}k\sqrt{k}+\sqrt{k+1}$$after which I'm stuck. In Unity, are there any pre-defined shortcuts, e.g. to open the dash or unhide the launcher? Footnote, non-inline syntax? I would like to use footnotes without cluttering the container paragraph. Something like:  This is a very long complex\footnote[a] sentence.  \foottext[a]{So complex that I would prefer to pull out the footnote text.}   The idea comes from the [linktext][1], [1]: http://url.com markdown syntax. I am from Nigeria and I'm doing a one year Internship program with The Marriott hotel in Wisconsin, I applied for the UK visitor visa two weeks ago to visit my fiance who lives and work in the UK, but I was refused the visa because the letter of my workplace that I provided was back dated of 2014 which they say they are not satisfy it accurately reflects my current employment circumstance. I also provided my pay slips, but they said they were not satisfied with it demonstrating the terms and conditions of my employment, and they also said I did not present any evidence to satisfy me having sufficiently strong ties and commitments in my country of habitual residence or elsewhere to motivate my timely departure from the UK. They said the documents my fiance presented were OK.  How soon can I re-apply for a UK visa after getting a refusal? Is it possible to open directories as administrator? I want to have a root privileges on some directories. How can I do this? What does the "yield" keyword do? What is the use of the yield keyword in Python? What does it do? For example, I'm trying to understand this code1: def _get_child_candidates(self, distance, min_dist, max_dist):     if self._leftchild and distance - max_dist &lt; self._median:         yield self._leftchild     if self._rightchild and distance + max_dist &gt;= self._median:         yield self._rightchild    And this is the caller: result, candidates = [], [self] while candidates:     node = candidates.pop()     distance = node._get_dist(obj)     if distance &lt;= max_dist and distance &gt;= min_dist:         result.extend(node._values)     candidates.extend(node._get_child_candidates(distance, min_dist, max_dist)) return result  What happens when the method _get_child_candidates is called? Is a list returned? A single element? Is it called again? When will subsequent calls stop?   1. This piece of code was written by Jochen Schulz (jrschulz), who made a great Python library for metric spaces. This is the link to the complete source: [Module mspace][1]. I am an Indian passport holder travelling from China to India via Kuala Lumpur, on Air Asia tickets bought separately. The layover is 4 hours and 30 minutes.  Do I need a transit visa? about how to open a URL in your default browser using the Terminal. Basically, you use the command   xdg-open http://google.com   My default browser is Firefox. But lets say that I wanted to open a specific website in Chrome, instead of Firefox. Is there a command to open it in Chrome instead of Firefox? How to remove duplicate values from a multi-dimensional array in PHP In MySQL 5.0 why does the following error occur when trying to create a view with a subquery in the FROM clause?     ERROR 1349 (HY000): View's SELECT contains a subquery in the FROM clause   If this is a limitation of the MySQL engine, then why haven't they implemented this feature yet?  Also, what are some good workarounds for this limitation?  Are there any workarounds that work for any subquery in the FROM clause or are there some queries that can not be expressed without using a subquery in the FROM clause?    An example query (was buried in a comment):  SELECT temp.UserName  FROM (SELECT u1.name as UserName, COUNT(m1.UserFromId) as SentCount        FROM Message m1, User u1        WHERE u1.uid = m1.UserFromId        Group BY u1.name HAVING SentCount &gt; 3 ) as temp How many solutions $X^{10} - I=0$ has in $M_2(\mathbb{R})$? How many solutions $X^{10} - I=0$ has in $M_2(\mathbb{R})$? Where $M_2(\mathbb{R})$ denotes the set of $2 \times 2$ real matrices.  I absolutely have no idea of where I should start from. $I$ and $-I$ solve the equation, but are there any other non-trivial solutions? Are there infinitely many solutions? Tall curly brace I would like to put a tall (long) left curly bracket on the left side of the following:  \documentclass[12pt]{article}  \begin{document}  \noindent c: $\{$1, ..., n$\}$ $\rightarrow$ $\{$1, ..., n$\}$, such that  \hspace{20 mm}  \noindent c($a_i$) = $a_{i+1}$ for 1 $\le$ $i$ &lt; $l$  \hspace{20 mm}  \noindent c($a_l$) = $a_1$   \end{document}   Please let me know if you can help. Thanks! How can I use the instant messenger program Kik - most commonly installed on smartphones and other mobile devices - on a laptop or desktop running Win7Pro, without a phone being involved and without needing any kind of Google account?  The two ways of using Kik from my laptop that I have so far found are both inadequate, or at least I do not how to tweak them to make them meet the requirements.  The first is to install on my laptop the Bluestacks emulator of Google's Android operating system, and then to run Kik on it. Unfortunately to run Kik on Bluestacks requires (or at least I have not found a way to avoid) having a Google Play account.  The second is to run Bluestacks on Manymo, the online emulator of Google's Android OS. Unfortunately, using Kik on Manymo similarly requires a Google Play account.  I realise Kik was made for use on a mobile OS, but it is supposed to be cross-platform where mobile OSes are concerned, and therefore not to be tied into Google's mobile OS - or, as far as I know, into any other software supplied by Google or any procedure prescribed by that company, such as giving it personal information - so I am optimistic that someone will be able to advise on what I am asking for here.  I notice that Microsoft have made a , so there might be a way to get Kik to work on this without dealing with Google. I couldn't find anything online about this, though.