Generalizing Cauchy-Schwarz for more than two vectors
Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$
What are the options for a non-EU national who wants to extend his stay in Switzerland (totaling less than 90 days)? If a product of relatively prime integers is an $n$th power, then each is an $n$th power Compare classifiers based on AUROC or accuracy? I have a binary classification problem and I experiment different classifiers on it: I want to compare the classifiers. which one is a better measure AUC or accuracy? And why?  Raondom Forest: AUC: 0.828  Accuracy: 79.6667 %            SVM: AUC: 0.542  Accuracy: 85.6667 % I have a vector space $V$ over a field of characteristic $0$. If $M_1,\dots,M_k$ are proper subspaces of $V$, and $N$ is a subspace of $V$ such that $N\subseteq M_1\cup\cdots\cup M_k$, how can you tell $N\subseteq M_i$ for some $i$?  I was first trying to show it just in the case $N\subseteq M_1\cup M_2$, and hoping to extend it to finitely many $M_i$. If either of the $M_i$ contains the other, the claim follows, so I suppose neither $M_i$ contains the other.   In hopes of a contradiction, I suppose $N\not\subseteq M_1$ and $N\not\subseteq M_2$, Picking $x_1\in N\setminus M_1$ and $x_2\in N\setminus M_2$, I'd have $x_1+x_2\in N\subseteq M_1\cup M_2$. The only thing I could conclude was that actually $x_1,x_2\notin M_1$ and $x_1,x_2\notin M_2$, which seems like a dead end. What's the right way to do this? Intermediate ring between a field and an algebraic extension. This is an exercise in some textbooks.     Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field.   The trouble is really with the inverse of $r$, where $r\in R$. How to prove that $r^{-1}\in R$, in apparent lack of a characterization of $R$.  It occurred to me to use the smallest field containing $R$ ($R$ is easily shown to be an integral domain), that's the field of quotients, and proving that it's $R$ itself, but I don't really know how to proceed.  A not-too-weak, not-too-strong hint will be much appreciated.   Beware $ $ Readers seeking only hints should  beware that there is now a complete answer. I used Sound Juicer to rip a CD of audio for a language learning book so that I could listen to them on my Android. However, Sound Juicer seems to only have the option of numbering files without leading zeros. Like this:  track_1.mp3 track_10.mp3 track_11.mp3   This leads to some confused ordering on my music player. So, I want to add some zeros to the name, so that they're ordered properly, like this:  track_01.mp3 track_02.mp3 track_03.mp3   How do I accomplish this? I tried using GPRename, but while it has the ability to add numbers, it doesn't seem to have any options for adding leading zeros.  Is there a better program or something I can do at the command line? Limiting number of lines in textarea I'm looking for a javascript that can limit the number of lines (by line I mean some text ended by user pressing enter on the keyboard) the user is able to enter in textarea. I've found some solutions but they simply don't work or behave really weird. The best solution would be a jquery plugin that can do the work - something like , but it should be able to limit text line count not character count. What is the fastest algorithm for finding the natural logarithm of a big number? Solvability and reducibility of a polynomial in a "chain" of finite fields I am wondering what the difference between a definition and an axiom.  Isn't an axiom something what we define to be true?  For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the successor of any natural number. Why can't we define 0 to be the natural number such that it isn't the successor of any natural number?  In general, why is an axiom not a definition?   Are there cases where we can formulate definitions as axioms or vice versa? Java generics T vs Object How can I change the umask for one command only? How can I interactively execute a command in Linux (zsh, if it matters) with a different umask from the default, for one command only? Perhaps a combination of commands combined in a single line? The new umask should apply only to that command and return to its default value for the next command entered.  Ideally the command would be agnostic to the default umask in force before its entered (in other words, the default umask doesn't have to specified). $c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$ Let $c$ is complex-number satisfying :  $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$  So, how could i get $(3c^{100}+\frac{2}{c^{100}}+1)(c^{100}+\frac{2}{c^{100}}+3)$ ? Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$ Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number.  The $A(p,q)$'s are known as alternating .     Can someone provide a nice proof that    $$A(1,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2?$$   I worked for a while on this today but was unsuccessful.  Summation by parts, swapping the order of summation, and approximating $H_k$ by $\log k$ were my best ideas, but I could not get any of them to work.  (Perhaps someone else can?)  I would like a nice proof in order to complete .  Bonus points for proving $A(1,2) = \frac{5}{8} \zeta(3)$ and $A(2,1) =  \zeta(3) - \frac{1}{2}\zeta(2) \log 2$, as those are the other two alternating Euler sums needed to complete my answer.   Added: I'm going to change the accepted answer to robjohn's $A(1,1)$ calculation as a proxy for the three answers he gave here.  Notwithstanding the other great answers (especially the currently most-upvoted one, the one I first accepted), robjohn's approach is the one I was originally trying.  I am pleased to see that it can be used to do the $A(1,1)$, $A(1,2)$, and $A(2,1)$ derivations. What are the left and right ideals of matrix ring? How about the two sided ideals? What are the left and right ideals of matrix ring? How about the two sided ideals? How did Montague Apparate into the Hogwarts bathroom? In year 5, Fred and George jam Montague into the Vanishing Cabinet. He's found weeks later in a toilet.      ‘He never managed to get all the words out,’ said Fred, ‘due to   the fact that we forced him head-first into that Vanishing Cabinet   on the first floor.’      […]      ‘They’ve found Montague, sir, he’s turned up jammed inside a toilet   on the fourth floor.’   ‘How did he get in there?’ demanded Snape.   ‘I don’t know, sir, he’s a bit confused.’   In year 6 Malfoy expands on that.      ‘I had to mend that broken Vanishing Cabinet that no one’s used for years. The one Montague got lost in last year.’      […]      ‘The other’s in Borgin and Burkes,’ said Malfoy, ‘and they make a kind of passage between them. Montague told me that when he was stuck in the Hogwarts one, he was trapped in limbo but sometimes he could hear what was going on at school, and sometimes what was going on in the shop, as if the Cabinet was travelling between them, but he couldn't make anyone hear him ... in the end he managed to Apparate out, even though he’d never passed his test. He nearly died doing it.   What happened here? Did he Apparate out into Hogwarts, Apparate into Borgin and Burkes and somehow wander back to Hogwarts and get stuck into a toilet? Recently I have installed an older version of Ubuntu on my old machine. Whenever I try to install any software, I get an error saying it couldn't be found:  $ sudo apt-get install vlc Reading package lists... Done                Building dependency tree        Reading state information... Done     E: Couldn't find package vlc Ubuntu 16.04 sftp and vscode ssh not working Implementation of "show" for function Does a word rhyme with itself?