Let $p$ and $q$ be linearly independent vectors. How can I prove that the matrix  $$pq^T +qp^T$$  always has rank $2$? I've tried several experiments, but still cannot prove it.
Let $p, q \in \mathbb{R}^n$ such that $\|p\| = \|q\| = 1$ and define $$A = pq^T + qp^T$$ Find the null space of $A$.    I am not having very much luck. I have managed to show that $p + q$ and $p - q$ are eigenvectors, but I can't seem to connect those two. I know that since $A$ is generated by an outer product, it is of rank one. That implies that the rank of the null space is $n - 1$, but I can't seem to get any further. Can anyone help?
How to establish the following identities without the help of calculus:  For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$  and  $$\sum_{0\le r\le n}\frac{(-1)^r\binom nr}{4r+1}=\frac{4^n\cdot  n!}{1\cdot5\cdot9\cdots(4n+1)}$$ How to reset postgres' primary key sequence when it falls out of sync? I ran into the problem that my primary key sequence is not in sync with my table rows.   That is, when I insert a new row I get a duplicate key error because the sequence implied in the serial datatype returns a number that already exists.  It seems to be caused by import/restores not maintaining the sequence properly. Puzzling Enumerable.Cast InvalidCastException The following throws an InvalidCastException.  IEnumerable&lt;int&gt; list = new List&lt;int&gt;() { 1 }; IEnumerable&lt;long&gt; castedList = list.Cast&lt;long&gt;(); Console.WriteLine(castedList.First());   Why?  I'm using Visual Studio 2008 SP1. Where can questions about our community-run moderation tools be asked? The question  was recently closed due to not being "about the software that powers the Stack Exchange network, within the scope defined in the help center." MEE is Johann Gambolputty made a comment that     SmokeDetector is a community run project to help to fight spam. Therefore your question is unfortunately off-topic for this site. You can find help in  chatroom.   If questions about community run moderation bots (e.g. SmokeDetector, FireAlarm, Natty, Queen, etc.) are off-topic here on Meta Stack Exchange, is there anywhere on the Stack Exchange network where one can post a Question and Answer-style question on them, or are they only on-topic in appropriate chat rooms? Extract date from a variable in a different format Let me explain you the problem   $ date +%c -d "$d" Tue 31 Dec 2013 01:13:06 PM CET $ date +'Today is %F' -d "$d" Today is 2013-12-31   This solution corresponds to current date.  But I have one variable which stores date other than current date  $Prev_date="Wed Dec 25 06:35:02 EST 2013"    I am looking for solution to read this date as 2013-12-25 and store it in a variable.  I have tried this:  a=`date --date=$Prev_date '+%y/%m/d'` echo $a   It's giving this error:  date: illegal option -- date=Wed usage: date [-u] mmddHHMM[[cc]yy][.SS] date [-u] [+format] date -a [-]sss[.fff] Suppose $f,g$: $\Bbb R$ → $\Bbb R$ are continuous functions such that $f(r) = g(r)$ for all $r$ in $\Bbb Q$. That is to say, $f$ and $g$ are equal on the rational numbers.   Prove that  $f(x) = g(x)$  for all $x$ in $\Bbb R$ Where is Jon going? At the end of S08E06 of Game of Thrones, Jon Snow along with Tormund and others of the Night's Watch      leave the Night's Watch.   and go into the woods.  Where are they going? And why? Ubuntu 16.04 software center doesn't have many apps I'm just starting to learn LaTeX. I tried typing \begin{equation*}  (math goes here)  \end{equation*}  and I got a whole bunch of errors. It seems LaTeX interpreted the empty line as the equation: when it saw nothing there, I guess it just ended math mode? It then said my other commands needed $'s so I assume that's what happened. When I typed \begin{equation*} (math goes here) \end{equation*}  everything was fine. So I'm confused. If LaTeX sees an empty line in an environment, does it just interpret this as me changing my mind, and deciding not to use the declared environment? How to solve a system of PDEs with zip condition? How I can create a graph of acceleration and speed - tikz? How can I create a graph of acceleration and speed like the one in the picture. Knowing that they are each other's consequence?    \documentclass{article} \usepackage{tikz}  \begin{document}  \begin{tikzpicture}               \end{tikzpicture} \end{document} How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number? Is my baby's gender an example of Schrodinger's cat? How to get the length of longest string in an array Why does light bend towards the normal when passing from a rarer to a denser medium? Whenever light rays are entering a denser medium they bend towards the normal.  Why do rays choose the path towards the normal? Why cannot they choose the path away form the normal? How do you get \mathbb{1} to work (characteristic function of a set)? I am trying to get the digit one printed the \mathbb{} font in math mode, however the command  \mathbb{1}   outputs some rubbish. I certainly saw the character I am after in some pdf's generated from latex, hence my question: how do you get it to work? If angular velocity & angular acceleration are vectors, why not angular displacement? Are angular quantities vector?   ... It is not easy to get used to representing angular quantities as vectors. We instinctively expect that something should be moving along the direction of a vector. That is not the case here. Instead, something(the rigid body) is rotating around the direction of the vector. In the world of pure rotation,a vector defines an axis of rotation, not a direction in which something moves.   This is what my book (authored by Halliday, Walker, Resnick) says. But they mentioned with a caution that      Angular displacements cannot be treated as vectors. ....to be represented as a vector, a quantity must also obey the rules of vector addition..... Angular displacements fail this test.   Why do they fail in this test? What is the reason? How Do You Use An Image As The World Background? Not all files are visible over MTP I have LG L5 (e610 model) device with Cyanogenmod 10. When I connect the phone to a computer with Linux (I have mtpfs installed), I can see both internal storage and the SD card.  However when I navigate inside them, I can't see some of the files.  For example, if I open file manager app on the phone, I can see several files in the Internal storage/Notifications directory, however the directory on the computer only shows one of those files.  If I switch USB connection type to PTP (camera) - as OS X doesn't support MTP natively - Image Capture sees the phone as a camera, however it can only see about a third of all the images present on the phone.  I don't have a Windows machine to try, but I suspect the results would be similar.  I tried upgrading Cyanogenmod to the latest nightly, clearing and reinstalling it - same results.  Any idea as to what is going on?  How can I make all files appear on the host computer? Deriving the Coulomb force equation from the idea of photon exchange? Since Newton's law of gravitation can be gotten out of Einstein's field equatons as an approximation, I was wondering whether the same applies for the electromagnetic force being the exchange of photons. Is there an equation governing the force from the exchange of photons? Are there any links which would show how the Coulomb force comes out of the equations for photon exchange? I know that my question is somewhat similar to the one posted here , but it doesn't really have an answer to my question specifically.