Let A be a real square matrix and $I$ the identity matrix of the same size. Let n > 0 be the smallest natural number for which $a_0, a_1, \ldots, a_{n-1} \in \mathbb{R}$ exist such that the following is true:  $A^n+a_{n-1}A^{n-1}+ \ldots +a_1A +a_0I = O$  Prove that A is invertible if and only if $a_0 \ne 0$ .
Let $A$ be a real square matrix en let $I$ be the identity matrix with the same size. Let $n&gt;0$ be the smallest natural number for which there exist  $a_0,a_1,...a_{n-1} \in {R}$ such that $A^n+a_{n-1}A^{n-1}+....+a_1A+a_0I=O$.  Show that A is invertible if and only if $a_0\neq0 $.  So i need to proof    $A$ invertible $\rightarrow$$a_0 \neq 0$ $a_0 \neq0 \rightarrow$ $A$ invertible    I don't really know where to start with 1, and with 2 I tried to prove it with a contradiction but I don't think it's right either.
Can I pickle a python dictionary into a sqlite3 text field? Any gotchas I should be aware of?  Can I store it in a text field, or do I need to use a blob? (I'm not overly familiar with either pickle or sqlite, so I wanted to make sure I'm barking up the right tree with some of my high-level design ideas.) How to concatenate a variable number of csv, removing their header rows? Is it possible to get a decimal output from doing division in Bash? Basically in Bash, what I'm trying to do is take a input of seconds from a user and find the hours. So basically if the user enter 35 the output should be 0.00972222.  But bash gives me is a zero.  heres the command I have:  echo "Enter the seconds you wish to convert to hours: " &amp;&amp; read sec &amp;&amp; echo " $((sec/3600)) is the amount of hours "   Is there a way I can make it so it prints out 0.00972222 if I enter 35.  Thanks! How does the 2009 Star Trek film mesh with Enterprise canon? All of the events in the 2009 Star Trek movie would have taken place after the adventures of Captain Johnathan Archer and company.  So, unless J.J. Abrams and friends decide to completely hand-wave that bit of history as well, all of the events in Enterprise should (in theory) still be able to line up with this new timeline.  Are there any points of conflict in this?  How about any time-travel episodes in the Enterprise series which may have happened differently after the interventions of Nero?  Are there possible good explanations for such conflicts, or are there some which are irreconcilable?  NOTE:  I'm not arguing here whether or not ST2k9 is a part of Star Trek canon.  In fact, this question is specifically granting that it is.  I just want to know if there's any conflicts between the new movie and the existing Star Trek history that remains applicable to it. Clear 301-redirect cache in Firefox Is it possible to clear only 301-redirects from Firefox's cache? CTRL+F5 doesn't do the job since you just reload the target without caching but do not clear the cached redirect.  An option to disable caching of those redirects completely would be fine, too - but some way (possibly through an extension) to remove those cached redirects would be better.  Note: I'm not looking for answers that disable all caching (including CSS, JS, etc.) Integrate $\int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$ Which is correct “Is that yours?” or “Is that your’s?”?  I ask because it is possessive, so I would think it would be the latter, but I typically use and see the former usage.  Are there particular cases in which one should be used instead of the other? Or is one simply correct and the other not?  This is one of the few things that still confuse me, so help is greatly appreciated. Can Java's NullPointerException be changed to report on which variable was null? How to read "service --status-all" results I need to identify the currently running services but I can't figure out the results of running service --status-all, I mean what does ?, - and + mean ?  $ service --status-all  [ + ]  acpid  [ + ]  anacron  [ + ]  apparmor  [ ? ]  apport  [ ? ]  atieventsd  [ + ]  avahi-daemon  [ ? ]  binfmt-support  [ + ]  bluetooth  [ - ]  brltty  [ + ]  console-font  [ + ]  console-setup  [ + ]  cron  [ + ]  cups  [ + ]  cups-browsed  [ - ]  dbus  [ ? ]  dns-clean  [ + ]  friendly-recovery  [ - ]  grub-common  [ ? ]  irqbalance  [ - ]  kerneloops  [ ? ]  killprocs  [ + ]  kmod  [ ? ]  lightdm  [ - ]  lm-sensors  [ ? ]  mysql  [ ? ]  networking  [ ? ]  ondemand  [ ? ]  pppd-dns  [ - ]  procps  [ - ]  pulseaudio  [ ? ]  rc.local  [ + ]  resolvconf  [ + ]  rfkill-restore  [ + ]  rfkill-store  [ - ]  rsync  [ + ]  rsyslog  [ + ]  saned  [ ? ]  sendsigs  [ + ]  setvtrgb  [ ? ]  speech-dispatcher  [ - ]  sudo  [ + ]  timidity  [ + ]  udev  [ ? ]  umountfs  [ ? ]  umountnfs.sh  [ ? ]  umountroot  [ - ]  unattended-upgrades  [ - ]  urandom  [ + ]  virtualbox  [ - ]  x11-common   And would running this command with sudo make any difference ? I tried it and in my case it didn't make any difference, but may it differ in other setups ? If you cast Polymorph on a non beast creature then cast Awaken on their beast form how much carries over after polymorph ends?  Polymorph does not give any explicit information on what happens to your statistics when the polymorph ends:     The transformation lasts for the duration, or [drops to 0 hp]. [...] The target's game statistics, including mental ability scores, are replaced by the statistics of the chosen beast.  (Basic Rules, pg. 266)   So this leaves ambiguity when game statistics are altered during polymorph,   The charm effect of Awaken seems pretty clear from the  and reinforced by the specific wording referring to the target as a beast or plant during effects and a creature afterwards:     The awakened beast or plant is charmed by you [longer than polymorph.] When the charmed condition ends, the awakened creature chooses [...] (Basic Rules, pg. 216)   The other two effects of awakening (other than making a plant mobile) are less clear-cut:     The target gains an Intelligence of 10. The target also gains the ability to speak one language you know.  (Basic Rules, pg. 216)   I would guess the modification to intelligence would apply to the beast's stats and therefore end with polymorph.     as being independent of creature statistics and act more as the character's experience. I'd argue the experience of a stranger turning you into a mouse, then gifting you the ability - as a mouse - to speak a language you have never used before would be an experience you couldn't lose I'm not sure if the language and intelligence granted from awakening should remain linked.    These are two examples I can see this having significant impact on my campaign:   Meet an NPC I don't share a language with, Polymorph them then Awaken their beast form so they can speak my language. Can I now communicate with them when they turn back?  If their original intelligence was higher then 10 is it reduced to 10? Say I use find familiar to get an owl (classified as a fey) familiar, then polymorph it to an owl (classified as a beast) so my druid friend can cast Awaken on it, does my familiar keep an intelligence of 10 and/or the ability to speak?   So after reverting from an awoken polymorphed state would you keep the 10 intelligence and a new language, only the new language, or not keep either? How to make a declaration from a command? There are two forms of setting the type style: the declarative method e.g, \itshape and the command form, e.g, \textit. The command form takes an argument whereas the declarative form does not --- it just applies itself to whatever comes afterwards.  I would like to develop a declarative form, say \UppercaseIt, of the command form \MakeUppercase{&lt;text&gt;}. I have read, in source2e.pdf on CTAN the code for implementing \textit, \itshape and \MakeUppercase and can't claim that I understand any of them. I am after something like {\UppercaseIt some text} instead of \MakeUppercase{some text}.  I only have a code outline of what I'm after, since I don't know how to do it:  \documentclass{article}  \newcommand{\UppercaseIt}{%   %... magic }  \begin{document}  \MakeUppercase{some text}% Yields SOME TEXT  {\UppercaseIt some text}% Should yield SOME TEXT  \end{document} boolean variable values in PHP to javascript implementation When has Wonder Woman died? I know Wonder Woman has died in canon before, but in which issue did this happen? I tried to install OracleJDK 8 using   sudo add-apt-repository ppa:webupd8team/java   But it's not working.   I searched the problem and found that      "The Oracle JDK License has changed for releases starting April 16, 2019." And  "Oracle Java (JDK) 8 Installer PPA (DISCONTINUED)"   So I installed the OpenJDK8 .   But why oracle cancelled the license of oracle 8 installer? Product of elements of a finite abelian group Suppose $G=\{a_1,...,a_n\}$ is a finite abelian group, and let $x=a_1a_2\dotsm a_n$.   Prove that if there is more than one element of order $2$ then $x=e$.   What I've done so far: (#1 is just for illustration, There should be at least $3$ such elements)   $G$ has an even number of elements with order $&gt;2$ (They are paired so: $(a,a^{-1})$ ). And then there are $e$ and $\{b|o(b)=2\}$. Also, if $G$ has at least one element of order 2 then $|G|$ is even (Lagrange). Thus $|\{b|o(b)=2\}|$ must be odd. If there is exactly one element $b$ of order $2$ then $x=b$. That's because $G$ is abelian, and we can write: $x=e\cdot b\cdot(a_1 a_{1}^{-1}\dotsm a_n a_{n}^{-1})=b\cdot(e\dotsm e)=b$. If there are exactly $3$ such elements $a,b,c\in G$, then, as shown above, $x=abc$ and thus $x=e$ (That's because $(abc)^2=a^2b^2c^2=e$ which means $abc\in \{a,b,c,e\}$. If $abc=a$ then $b=c^{-1}=c$ which is false, same goes for $abc=b,abc=c$.)   I haven't managed to find a reason why the claim must hold for $5, 7, ...$ (i.e. All other odd integers). Does anyone have an idea?  Thanks in advance. Please excuse my English.  edit: Not a canonical answer but a simple one (if such even exists). In a particular textbook, the work function of a metal (in the context of the photoelectric effect) is defined as:     the minimum amount of energy necessary to remove a free electron from the surface of the metal   This sounds similar to ionisation energy, which is:     the amount of energy required to remove an electron from an atom or molecule in the gaseous state   These two energies are generally different. For instance, Copper has a work function of about 4.7eV but has a higher ionisation energy of about 746kJ mol-1 or 7.7eV.  I've sort of figured it's because the work function deals with free electrons whilst ionisation is done with a valence electron still bound within the atom. Is the difference due to the energy required to overcome the attraction of the positive nucleus? How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer let $a_{i},i=1,2,\cdots,n$ be postive integer ,show that $$1^{n-1}2^{n-2}\cdots (n-2)^2(n-1)|\prod_{1\le i&lt;j\le n}(a_{i}-a_{j})$$  I know this $\prod_{1\le i&lt;j\le n}(a_{i}-a_{j})$ is  Vandermonde  determinants,and I found $$1^{n-1}2^{n-2}\cdots (n-2)^2\cdot (n-1)=1!2!3!\cdots (n-1)!=\prod_{1\le i&lt;j\le n}(j-i)$$ we only prove $$\prod_{1\le i&lt;j\le n}\dfrac{a_{j}-a_{i}}{j-i}$$ is integer  maybe consider Vandermonde determinants ? But I can't prove this Print two files in two columns I want to print two files in two columns -- first file on the left side and second on the right side. paste doesn't do the job, because it can only insert a character as delimiter, so if first file lines have different length output will be twisted: $ cat file1 looooooooong line line $ cat file2 hello world $ paste file1 file2 looooooooong line   hello line    world  If it was a command to add trailing spaces like fmt --add-spaces --width 50 the problem would be solved: $ paste &lt;(fmt --add-spaces --width 50 file1) file2 looooooooong line                                 hello line                                              world  But I don't know a simple way to do this. So how to merge and print several files horizontally without twisting? Actually, I just want to look at them simultaneously.  UPD: command to add trailing spaces does exist (for example, xargs -d '\n' printf '%-50s\n') But solution like $ paste &lt;(add-trailing-spaces file1) file2  does not work as expected when file1 has fewer lines than file2. I'm creating a composite image, in order to create a timeseries. I've already created the composited image, but i am having difficulty masking the clouds of the composite image. I've followed the code from a page having similar problems, however when I run the code (after replacing all the l8_mayo to composite as that is the name of my composite var), I get the error code of composite.map not a function. This error does not come up when I run the exact same code from the previous answer. Can someone explain how to overcome this problem? Here is the page I used:  here is my code and the link to my code:  (point is the long, lat for Lake Manyara, Tanzania)  //load images for composite var sr14= ee.ImageCollection('LANDSAT/LC08/C01/T1_SR') .filterBounds(point) .filterDate('2014-01-01','2014-03-01') .select(['B4','B3','B2']);  // Temporally composite the images with a maximum value function. var composite = sr14.max(); Map.setCenter(35.8244, -3.5799); Map.addLayer(composite, {}, 'max value composite');  var getQABits = function(image, start, end, newName) { // Compute the bits we need to extract. var pattern = 0; for (var i = start; i &lt;= end; i++) {    pattern += Math.pow(2, i); } // Return a single band image of the extracted QA bits, giving the band // a new name. return image.select([0], [newName])               .bitwiseAnd(pattern)               .rightShift(start); };  // A function to mask out cloudy pixels. var cloud_shadows = function(image) { // Select the QA band. var QA = image.select(['BQA']); // Get the internal_cloud_algorithm_flag bit. return getQABits(QA, 7,8, 'Cloud_shadows').eq(1); // Return an image masking out cloudy areas. };  // A function to mask out cloudy pixels. var clouds = function(image) { // Select the QA band. var QA = image.select(['BQA']); // Get the internal_cloud_algorithm_flag bit. return getQABits(QA, 4,4, 'Cloud').eq(0); // Return an image masking out cloudy areas. };  var maskClouds = function(image) { var cs = cloud_shadows(image); var c = clouds(image); image = image.updateMask(cs); return image.updateMask(c); };  var composite_free = composite.map(maskClouds);  Map.addLayer(composite_free, 'composite collection without clouds'); Did Luke attempt to retake the Jedi Temple on Coruscant? After the New Republic gains control of Coruscant through the Galactic Concordance, did Luke Skywalker attempt to reclaim (or even visit) the former and still very much intact Jedi Temple, now converted to be the deceased Emperor's palace?   If Luke is so insistent on saving a tree just because it belongs to the Jedi and the Empire stole it, then shouldn't reclaiming the Jedi Temple be even higher priority?  If he did, is he aware of the Sith shrine beneath it? Which is something even Yoda does not know. And if he didn't, why didn't he?