let $V$ be a  euclidean space. $T:V\to V$ be an orthogonal  linear transformation. $W\subset V$ is a $T$-invariant subspace.  I need to prove 2 things:  A. $T\bigl|_W$ is orthogonal  so I said that if $T$ is an orthogonal  linear transformation, then it preserves inner products and for every $v,u$ that belong to $V$, $\langle\, Tv,Tu\,\rangle$ = $\langle\, u,v\,\rangle$.   so if $W\subset V$, then for every $w_1, w_2$ belonging to $W$,  $\langle\,Tw_1,Tw_2\,\rangle=\langle\, w_1,w_2\,\rangle$.  but it feels wrong, I'm not sure whether it is right.  B. Show that $W^\perp$ is $T$-invariant. which I have no idea.
Here is a question as phrased in Advanced Linear Algebra (Steven Roman):     Show that a linear operator $\tau$ on a finite-dimensional complex inner product space $V$ is normal if and only if whenever $S$ is an invariant subspace under $\tau$, so is $S^\perp$.   I'm only interested in the $\Rightarrow$ direction. Wikipedia (and ProofWiki)  that uses some kind of computation with the trace. Linear Algebra Done Right chooses an orthonormal basis and then uses the fact that $\Vert \tau v \Vert = \Vert \tau^* v \Vert$.  However, the above  proofs seem a bit too computational to me. Can the following argument work?  (Use $\odot$ for orthogonal direct sum.) Suppose that $\tau$ is normal and write $$V=E_{\lambda_1} \odot \cdots \odot E_{\lambda_k}$$ where the $E_{\lambda_i}$ are the eigenspaces of $\tau$. If $S$ is a $\tau$-invariant subspace then $\tau|_S$ is diagonalizable, so $$S=\widetilde{E}_{\lambda_1} \odot\cdots\odot \widetilde{E}_{\lambda_k}$$ where the $\widetilde{E}_{\lambda_i} \le E_{\lambda_i}$ are the eigenspaces of $\tau|_S$ (which could be trivial). Then choose orthogonal complements $D_i$ such that $E_{\lambda_i} = \widetilde{E}_{\lambda_i} \odot D_i$, and then $$S^\perp = D_1 \odot\cdots\odot D_k.$$ Since each $D_i$ is a subspace of an eigenspace, they are all $\tau$-invariant.
Making holes in a mesh I'm trying to install programs/files from the getdeb repo but get a "check your internet connection" error.  Programs than I've tried include Qbittorrent and Mozilla-Plugin-VLC.    Through the terminal I get the following errors:  Err http://archive.getdeb.net/ubuntu/ natty-getdeb/apps qbittorrent amd64 2.8.2-1~getdeb2   Could not connect to archive.getdeb.net:80 (209.105.191.78). - connect (113: No route to host) E: Failed to fetch http://archive.getdeb.net/ubuntu/pool/apps/q/qbittorrent/qbittorrent_2.8.2-1~getdeb2_amd64.deb: Could not connect to archive.getdeb.net:80 (209.105.191.78). - connect (113: No route to host)   &amp;  Err http://archive.getdeb.net/ubuntu/ natty-getdeb/apps mozilla-plugin-vlc amd64 1.1.10-1~getdeb1   Could not connect to archive.getdeb.net:80 (209.105.191.78). - connect (113: No route to host) E: Failed to fetch http://archive.getdeb.net/ubuntu/pool/apps/v/vlc/mozilla-plugin-vlc_1.1.10-1~getdeb1_amd64.deb: Could not connect to archive.getdeb.net:80 (209.105.191.78). - connect (113: No route to host)   I'm not having any problems adding/removing other programs/files from my machine.  Is there a problem with GETDEB repo (and if so, is there any way to fix it on my end) or is there something wrong with my computer's configuration? Let $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ a long of $Y \subset X$, with exceptional divisor $E$ and $\text{dim}Y &gt; 0$,   where $X$ and $Y$ are smooth projectives schemes.   For an answer to a question asked here in the forum (), Blowing up and direct image of line bundle), we have: $$\pi_{*}\mathcal{O}_{\widetilde{X}}(-E) = I_{Y/X} \tag{$**$}$$ here $I_{_{Y/X}}$ denotes the ideal sheaf of $Y$.  $(**)$ was obtained through the following exact sequence: $$0 \longrightarrow \mathcal{O}_{\widetilde{X}}(-E) \longrightarrow \mathcal{O}_{\widetilde{X}} \longrightarrow \mathcal{O}_{E} \longrightarrow  0$$  My doubt is: How to find $\pi_{*}\mathcal{O}_{\widetilde{X}}(-nE)$ for $n \geq 0$? What if $n &lt; 0 $?  Thank you. How does the SQL injection from the "Bobby Tables" XKCD comic work? Just looking at:   (Source: )  What does this SQL do:  Robert'); DROP TABLE STUDENTS; --   I know both ' and -- are for comments, but doesn't the word DROP get commented as well since it is part of the same line? Why is $\frac{1}{2\pi i}\int_{\gamma}\frac{f'(z)}{f(z)}dz$ an integer? Let $f$ be holomorphic in an open $\Omega \subset \mathbb{C}$ and $\gamma$ a closed curve in $\text{int}(\Omega)$, along which $f$ is never zero.   Are these hypotheses enough to claim $\frac{1}{2\pi i}\int_{\gamma}\frac{f'(z)}{f(z)}dz$ is an integer? If not, what are the necessary and sufficient conditions for that?  I can prove that's an integer for some particular cases, for example when $\Omega$ is convex and  $\gamma$ is a circle. But I've seen people claming this in many other contexts, like when $\Omega$ is an annulus around $0$ and $\gamma $ is an arbitrary curve inside it. This isn't obvious to me at all.  Thanks! The problem from Burton: show that the equation $n^2 + (n+1)^2 = m^3$ has no solution in the positive integers.  So far, I can see that gcd($n$,$n+1$)$=1$ and $m \equiv_4 1$ and $m=a^2 + b^2$ for some integers a,b. I'm guessing I need to reach a contradiction.  At this point, I am stuck. Any hints? Is it possible to export the equations from Mathematica to MATLAB? Is "You are correct" correct? I need to make my (BibTeX) references section appear in the table of contents of my LaTeX document (documentclass: article), with section numbering too.  My approach until now has been making a new section and including the bibliography (references.bib) at that point:  \section{References} \bibliography{references}   However, the final document shows both the section title that I have written and the section title that BibTeX writes, which is quite redundant and I definitely dislike.  How can I either remove BibTeX's section title, or make the BibTeX bibliography appear in the table of contents without making a new section?  If I were to make the BibTeX bibliography appear in the table of contents without making a new section, how could I assure that the section title that BibTeX writes looks exactly like sections typeset with \section? What is the best way to get the names of all of the tables in a specific database on SQL Server? Is there any danger to letting food cook in a slow cooker for a very long time? I've recently started using a slow cooker and I was thinking about recipes that could be cooked for days.  My main concern is if there are any side effects for leaving food cooking for days that would make this plan unfeasible (dangerous to try).  Since it is at cooking temperature, there wouldn't be any reason to worry about bacteria growing on the food.  Liquid would be lost over time, which could result in burning if left unchecked, but if I add more every morning/night as needed, that too wouldn't be a problem.  All I can see is that most recipes wouldn't end up tasting as well, but undesirable isn't equal to dangerous.  There just seems to be something absurd with the notion of leaving food cooking for 5 days or even 2 weeks without there being any dangers, but I'm having a hard time figuring out what they would be. Length of generator output Python provides a nice method for getting length of an eager iterable, len(x) that is. But I couldn't find anything similar for lazy iterables represented by generator comprehensions and functions. Of course, it is not hard to write something like:  def iterlen(x):   n = 0   try:     while True:       next(x)       n += 1   except StopIteration: pass   return n   But I can't get rid of a feeling that I'm reimplementing a bicycle.  (While I was typing the function, a thought struck my mind: maybe there really is no such function, because it "destroys" its argument. Not an issue for my case, though).  P.S.: concerning the first answers - yes, something like len(list(x)) would work too, but that drastically increases the usage of memory.  P.P.S.: re-checked... Disregard the P.S., seems I made a mistake while trying that, it works fine. Sorry for the trouble. Behaviour of return statement in catch and finally Is there a simple reason why the expected number of coin flips till getting $m$ more heads than tails or $n$ more tails than heads should be $mn$? I flip a coin until I get $m$ more heads than tails, or $n$ more tails than heads. Let the expected number of flips of the coin before stopping be $f(m,n)$.  I obtained $f(m,n)=mn$ from the recursion $f(m,n)=1+\frac{f(m-1,n+1)+f(m+1,n-1)}2$ with $f(k,0)=f(0,k)=0$ for all $k$.   Other than going through this recursion (and either solving by inspection or by writing as linear recurrence in single variable and solving brute force), is there an intuitive reason you should expect this process to take $mn$ flips? I was thinking about the more general problem with probability $p$ of getting heads and was struck by how simple the formula became when handling, what turned out to be a special case (general formula broke down) of $p=\frac12$. Commenting out large sections Closest points on two line segments I am looking for a general formulation to find the closest points on two line segments. What I was thinking about is to define our lines as:  $$ P1 + s  (P2-P1)$$  $$ Q1 + t  (Q2-Q1)$$  Where $P1 , P2, Q1$ and $Q2$ are the beginning and the end points on each segment.  Now we should go through an optimization problem as: $\min f(s,t)$ such that $0&lt;s&lt;1$ and $0&lt;t&lt;1$.  Where $f(s,t)$ is the point-to-point distance function.  Is there any straight forward solution? At the end of The Return of the King, Frodo, Bilbo, Gandalf, Elrond, and Galadriel left Middle-earth on a ship, along with other elves. I named those five because they were all ringbearers and it seems to be a common trait. Frodo and Bilbo bore the One Ring and the three others bore the three elven Rings.  Why did they have to do that? Why didn't they stay and have a peaceful life now that the Dark Lord was gone? 90s (or earlier) Japanese anime movie: two planets at war, young man is in love with princess from the other planet, and floods his own village I would like to find out the title of a Japanese anime feature film (not a series) that I saw on Romanian TV in the early 90s.   It was about a feud between the peoples of two planets (one very technologically advanced, the other not). The two planets go to war against each other, and a young man from one of the planets falls in love with a princess from the other planet.  I recall a scene where the army of the second planet take off in hot air balloons when they go to war.  The only other thing I recall is that there is a dam next to the village where the young man lives, and for some reason he opens the dam gates and floods his own village.   I do not recall there being robots in the movie, however I remember very vaguely that the technology level was probably at the level of late 19th century Europe on one planet and pre-industrial on the other one.  What it's not:   Nobunaga the Fool Aldnoah Zero Queen Millennia Uchuu Senkan Yamato (movie) Context: I want to store the TIFF data to use it for running a linear regression model later on. Ultimate goal: Create a carbon map using field data as a basis. Libraries: library(sp) library(raster) library(tidyverse) library(rgdal)  plotcoord &lt;- SpatialPointsDataFrame(AGCplot[,3:4], AGCplot[,1:2],proj4string=CRS('+proj=longlat'))  pspcoord&lt;-spTransform(plotcoord, CRS('+proj=utm +zone=21')) #Omzetten naar UTM21N (= S1) plot(pspcoord)  s1 &lt;- stack('Some.tif')  colnamen = c(&quot;plotid&quot;,&quot;AGC&quot;,&quot;B2&quot;,&quot;B3&quot;,&quot;B4&quot;,&quot;B5&quot;,&quot;B6&quot;,&quot;B7&quot;,&quot;B8&quot;,&quot;B8A&quot;,&quot;B11&quot;,&quot;B12&quot;,&quot;NDVI&quot;,&quot;NDWI&quot;,&quot;MMRI&quot;,&quot;IRECI&quot;,&quot;MASI&quot;,&quot;VV&quot;,&quot;VH&quot;)  s1_e &lt;- extract(s1,pspcoord,method='bilinear')  Upon extraction I get the following message: Warning message: In proj4string(x) : CRS object has comment, which is lost in output How to derive the conjugate prior of an exponential family distribution I am trying to derive the conjugate prior of the univariate Gaussian distribution over both the mean and the precision. I know that the prior I'm looking for is the normal-gamma distribution, but the idea is to derive this result.  It seems that you can write down the conjugate prior of an exponential family distribution immediately if you represent it in the canonical exponential family form.  suggests a way to do just that. So far I have written the Gaussian probability function in this form and equated the following, but I don't see how the form of the conjugate prior is immediately obvious.  $$p(\mu, \lambda | D) \propto p(\mu,\lambda)p(D | \mu, \lambda)$$