Suppose $P$ and $Q$ are $n\times n$ matrices of real numbers such that   $P^2=P$ $Q^2=Q$ $I−P−Q$ is invertible, where $I$ is a $n\times n$ identity matrix.   Show that $P$ and $Q$ have the same rank.
Let $P,Q$ real $n\times n$ matrices  such that $P^2=P$ , $Q^2=Q$ and $I-P-Q$ is an invertible matrix.  Prove that $P$ and $Q$ have the same rank.  Some help with this please , happy year and thanks.
Which movie/TV show has an evil cyborg harvesting eyes from androids in a hidden town? I watched it maybe more than 8 to 12 years ago, but as far as I remember there was this place where androids (or maybe they were cyborgs) lived. (I think it was more like a hidden town). This guy finds them by accident (or at least I remember it that way) and tries to help them. I don't remember seeing any aliens or spaceships or anything like that. It was live-action, and I think US-based.  So, what I remember:   The bad guy is a human who has one red flashy robotic eye (not like terminators, it looks like a square eye patch above one of his eyes that functions like a second eye), he is collecting eyes from these androids/cyborgs. He also kills some of these androids/cyborgs but not all of them, some of them he just collects their eyes and doesn't kill (or get to kill). There was this woman who was a sexbot or something, she tried to give pleasure to the bad guy at some point, yet the bad guy added one of her eyes to his collection instead, but didn't/wasn't able to kill her. Towards the end she says something like "I used to give pleasure, now I only want to give pain" The human guy who found them tried to help them defend themselves from the bad guys, but they were in bad shape, many of them needed repair. To repair some of them, they needed spare parts they didn't seem to have, but one of the android/cyborg dudes said something like "we don't really bury our dead" and he brought a box of spare parts (or maybe android/cyborg body parts), they used these parts to repair some of the androids/cyborgs. There was this android/cyborg guy who was an assassin or something, but lost his fingers or hands, they couldn't find any hands compatible to his, but he said something like "I can still feel my trigger finger, just attach the weapons to my hands".   I remember it as a great movie/tv show, but I can't find anything about it. Can someone help me?  Please and thank you in advance... Suppose $X$ is a finite set and $f : X \to X$ is a function. Then $f$ is injective if and only if $f$ is surjective. Suppose $X$ is a set and $f : X \to X$ is a function.  If $X$ is a finite set, prove that $f$ is injective if and only if $f$ is surjective. Show that when $X$ is an infinite set, the statement is not true in general.   For that, let $X = \mathbb{R}$ and find a function $f: \mathbb{R} \to \mathbb{R}$ which is injective but not surjective, and a function $g: \mathbb{R} \to \mathbb{R}$ which is surjective but not injective  My attempt at the first part   Suppose that the function is surjective but not injective. Let $a,b \in X$ such that $f(a)=f(b)$ but $a \not =b$. Now since this is a finite set mapping to itself then there are not enough elements in $X$ to map to $X$ since two elements were used to map to one element, thus the function can't be surjective. Therefore if the function is surjective it is also injective.   Now suppose that the function is injective but not surjective. Let $a \in X$ such that $f(b)\not =a$ for all $b \in X$. Since the function was assusmed to be injective each $a \in X$ must map to an element of $X$, thus must have two elemetns in $X$ mapping to the same element in $X$. Hence the function can't be injective, therefore if the function is injective it must be surjective.   Would an exponetial function work for injective but not surjective since can't map to negative numbers. And an example of a function that is surjective but not injective could be a log function since none of the negative numbers get mapped to anything?   Also I'm having trouble seeing why it has to be finite for the first part to work. I understand it has something to do with the pigeonhole principle but its not that clear to me. Stack Overflow duplicated layer bug Originally posted on Stack Overflow:   Requirements to reproduce:  Windows 7 x64  Firefox 71.0 x64  Steps to reproduce:   Go to the ask question section Write something to the title field Start scrolling similar questions Change the title field Start scrolling the similar questions again Please help me  to evaluate combinatorially the following sum: $$\sum_{k=0}^n \binom{n}{k}$$   Thank you. How can I copy the path to a folder or a file? The Info Dialog doesn't allow me to copy the information that stands there ...    Is there an easy way to do this? PostGIS Geometry Query returns "Error: Operation on mixed SRID geometries" only for certain values Passive voice phrasing of "The bottle contains two liters" Connection pooling in PHP Is it possible to cache database connections when using PHP like you would in a J2EE container?  If so, how? The best free and most up to date phylogenetic tree on the internet? I found phylogenetic tree in wikipedia is lacks of accuracy. It actually confusing with versions. Some terms was not scientifically accurate with dna analysis anymore  So are there anyone know where is the most updated datasource on the internet? "How to format" block mentions syntax highlighting even if it is not available on the site When you edit the body of a post, the "How to Format" widget is shown in the right sidebar. It mentions the syntax highlighting function, even on  like :    It's probably also a good idea to move the part about code fences down for non-technical sites (like ); other formatting options are more important there. What does the typical workflow of a journal look like? Where is the line drawn with domain names which include a trademark? A search on google for "iphone developer" turns up loads of websites which have "iphone" in them, a trademarked name by Apple.  So I'm led to believe that having a domain such as iphonedeveloper.com is ok? Well, you're still using Apple's trademark, but it would be hard to brand yourself otherwise. You're an IPHONE DEVELOPER... right?  Well, what if I want to provide a website where users pay to get a list of the best offers from Ebay? I might have a domain like ebaydeals.com (I don't... i'm just speculating!).  Now I've heard that places like Ebay are really hot on the trigger and fire out emails to people who register domains like that straight away. But whats the difference?  In both cases I'm making money from the trademark, effectively, so is it just down to how lenient the company who owns the trademark is? Or are there rules? Is there a specific "line" you don't cross?  Thanks How to Mark NTFS filesystem dirty from *NIX/Linux? reflecting direct light on glossy surface in Cycles Working in Cycles.  Say I have a "disco ball" with a glossy shader. I'm pointing a spot light to it, and I'm trying to have the reflections of that light illuminate the diffuse material on the walls with the patterns from the ball.    I get this:    but I'd like to get something like:    For the illustration I placed a point light inside the ball an removed faces. But what I'd like to do is move the light around the ball so that the light reflected from it changes like it would do in real life.   Is that possible at this point?  EDIT:  Just for the record here's the image rendered in LuxRender (yet another program to learn...) Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively. Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$.   How to prove? I think these terms almost refer to the same thing, when used loosely:   terminal shell tty console   What exactly does each of these terms refer to? What does capital letter I mean in this formulas? Hello I don't understand meaning of the symbol represented by a capital letter "I" in the formulas below. There is  to the paper with the formulas.    At the beginning I thought it mean "where c is not equal to the result of classifier T(m)", however in step (d) there is a multiplication symbol before it, so it probably mean that "I" should "return" a value. Finding inverse of polynomial in a field I'm having trouble with the procedure to find an inverse of a polynomial in a field. For example, take:     In $\frac{\mathbb{Z}_3[x]}{m(x)}$, where $m(x) = x^3 + 2x +1$, find the inverse of $x^2 + 1$.   My understanding is that one needs to use the (Extended?) Euclidean Algorithm and Bezout's Identity. Here's what I currently have:  Proceeding with Euclid's algorithm:  $$ x^3 + 2x + 1  =(x^2 + 1)(x) + (x + 1) \\\\   x^2 + 1 = (x + 1)(2 + x) + 2$$  We stop here because 2 is invertible in $\mathbb{Z}_3[x]$. We rewrite it using a congruence:  $$(x+1)(2+x) \equiv 2 \mod{(x^2+1)}$$  I don't understand the high level concepts sufficiently well and I'm lost from here. Thoughts?  Wikipedia has a page on this we a , but it's still not clear in my mind.  Note that  has almost the same title, but it's a level of abstraction higher. It doesn't help me, as I don't understand the basic concepts.  Thanks. To translate a vector by 10 unit in the X direction, why do we have to use a matrix?    We can just add 10 to the mat[0][0], and we got the same result too. How do you show a hidden folder (whose name starts with .) in File Manager a.k.a. Nautilus on Ubuntu?  I am somewhat new to Ubuntu and can't seem to find the "view" menu.