Assume that $A\in\mathbb{R}^{n\times k}, \ B\in\mathbb{R}^{k\times m}$   and that $\text{Rank}(A)=r, \ \text{Rank}(B)=s.$ What can be said   about $\text{rank}(C),$ where $C=AB?$   The only thing I can quickly see here is that $C\in\mathbb{R}^{n\times m}.$ I also know that $\text{Rank}(C)=\text{Dim}(V(C)).$ But what is the dimension of $C$ then?  The exam has a really messy solution where the answers are that the rank of $C$ is sometimes $=s$ and sometimes $=r$. I don't understand any of it.
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you. Ring isomorphism (polynomials in one variable) Show all of my question/answers to me even if they are deleted Convert object string to JSON How do I know if I'm working on a Virtual Machine or not? Is there a way to know if the Windows machine I'm working on is virtual or physical? (I'm connecting with RDP to the machine. If it's a virtual machine it is working and handled by VMWare). How to add custom objects to the "add object" menu? How do I remap multi-key sequences to individual keys? The top bar is showing (again) the suggested edits count and 10K flags count, like it was before.  On Stack Overflow it looks like this: (that's the original color, same as in old top bar)        And on this very Meta site:        Same top bar. But different colors! Consistency broken.  I've just noticed the suggested edits count on Meta:        Black on black, I can see just a floating number - really confusing and out of any context.  In the name of Consistency, can we please have the same color on all sites? Each site got a &quot;hot&quot; tab, where its contents are calculated based on . However take a look on the screenshots below and see for yourself that &quot;hot&quot; got nothing to do with many of the questions listed there. :    :    As you can see, there are questions with negative score, no answers, and even closed. I won't be surprised to see spam in there as well. This can't be any worse. What I suggest, instead of fixing the formula or patching it, simply have the hot tab show the same questions as the Hot Network Questions sidebar. (from the specific site only, of course.) Since the hot questions &quot;pool&quot; (as seen ) is limited to 100 for the whole network, either expand it or have the per-site list show questions with less hotness points - it would still be better than what we have now. Crop jpeg into circular tikz node \documentclass[demo]{article} \usepackage{graphicx} \usepackage{tikz} \begin{document} \begin{tikzpicture} \node[circle,draw,inner sep=2cm]{\includegraphics{myimage.jpg}}; \end{tikzpicture} \end{document}   How do I modify the above code so that the included image is adapted to the circle shape and cropped of the exceeding parts?  I want to specify a shape (a circle) and then include an image respecting the given dimension and cropped. I have the following table:     \documentclass[a4paper,preprint,aps,                    unsortedaddress]{revtex4}     \usepackage[top=2.5cm,bottom=2.5cm,                 left=3.5cm,right=2.5cm]{geometry}     \usepackage{longtable,array,booktabs}      \ifdefined\mc        \renewcommand{\mc}{\multicolumn}     \else        \newcommand{\mc}{\multicolumn}     \fi      \begin{document}           \setlength\LTleft{-3.0cm}         \setlength\LTright{-2.0cm}          \scriptsize      \begin{longtable}{ l r @{--} p{1.3cm} &gt;{\centering}p{2.6cm}                         *{4}{p{1.25cm}} p{2.1cm}}                    %% headers and footers      \caption{Table} \\      \hline\hline     \endfirsthead      \mc{9}{c}{\tablename\ \thetable\ -- \emph{Continued from previous page}} \\      \hline     aaaaaaaaaaaaaaaaaaaaaaaa &amp; \mc{2}{c}{bbb--ccc} &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     \hline     \endhead      \hline     \mc{9}{r}{\emph{Continued on next page}} \\     \endfoot      \hline\hline     \endlastfoot      %% body of table          aaaaaaaaaaaaaaaaaaaaaaaa &amp; \mc{2}{c}{bbb--ccc} &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     aaaaaaaaaaaaaaaaaaaaaaaa &amp; bbb &amp; ccc &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     aaaaaaaaaaaaaaaaaaaaaaaa &amp; bbb &amp; ccc &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\     aaaaaaaaaaaaaaaaaaaaaaaa &amp; bbb &amp; ccc &amp; ddd &amp; eee &amp; fff &amp; ggg &amp; hhh &amp; iii \\      \end{longtable}     \end{document}   which I would like to shift to the left by setting \setlength\LTleft{-3.0cm}  and \setlength\LTright{-2.0cm}. Furthermore, I have to apply \scriptsize for the table. However, none of these commands are working with revtex4. Could you please provide a solution to this problem? (This question is the same as in , but I cannot sign up with that username.)  If it is not possible, an alternative solution would also be useful for me. For this case, I attached the first page of my real table. I currently have an Ubuntu-Windows Dual boot. However, I would like to try out a new OS so I burned Linux Mint to a USB stick.   I would like to install Linux Mint and remove Ubuntu while still keeping my windows dual boot. I've already backed up my Ubuntu setup so I can wipe it out.   In the Linux Mint installer, should I choosing the "install alongside windows" option do this? The Borel-Cantelli Lemma in  seems to be a sort of "corollary" of the .  It says:  "Let $(E_k)_{k=1}^{\infty}$ be a countable collection of measurable sets for which $\sum_{k=1}^{\infty} m(E_k) &lt; \infty$. Then almost all x $\epsilon \ \mathbb{R}$ belong to at most finitely many of the $E_k$'s"  I don't get three things:   "Then almost all x $\in \ \mathbb{R}$ belong to at most finitely many of the $E_k$'s"   What does that mean? Since the Borel-Cantelli Lemma comes after the "almost everywhere" definition, I guess: $\exists \ Z \ \subset \mathbb{R}$ s.t. $m(Z) = 0$ and $\forall x \ \in \ \mathbb{R} \setminus Z$, x belongs to $(E_{j}, E_{j+1}, E_{l}, ..., E_{m})$.   In the proof, one can see the $m(\limsup \ E_k) = 0$ part, stated as $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$. For some reason, $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$ implies that "Therefore almost all x $\in \ \mathbb{R}$ fail to belong to $\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]$ and therefore belong to at most finitely many $E_k$'s"   Okay, why? What is the contradiction if almost all $x \epsilon \ \mathbb{R}$ belong to all the $E_k$'s? Or a countably infinite subcollection?   What is the significance of this? Why not just state the Borel-Cantelli Lemma as "Let $(E_k)_{k=1}^{\infty}$ be a countable collection of measurable sets for which $\sum_{k=1}^{\infty} m(E_k) &lt; \infty$. $m(\cap_{n=1}^{\infty} [\cup_{k=n}^{\infty} E_k]) = 0$"? I'm basically searching for the opposite of putting all your eggs in one basket, where the risk is total failure because you did not hedge your efforts. I'm searching for a phrase that encompasses splitting your efforts and evokes the prospect of failing at both. A bird in the hand is worth two in the bush also encompasses a sense of having focused your efforts, but only alludes to total failure. To satisfy the criteria, the phrase must cover:  attempting two things (or multiple iterations of one thing) both of which you might succeed at, if you addressed them serially but both of which you fail at because you address them in parallel  Here are the idioms/phrases that do not sufficiently cover the idea and why:  &quot;A servant to two masters&quot;  This has the sense of competing priorities, but not the sense of failing to achieve your goals.  &quot;Jack of all trades. Master of none.&quot;  The sense is insufficiently pejorative. Being a jack-of-all-trades can be seen as an advantage; I'm trying to focus on the lack of mastery.  &quot;The man who fancied himself a wit, and was half-right.&quot;  From Christopher Hitchens. Hilarious, but lacking the sense of split effort between two things.  &quot;Walking and chewing gum at the same time.&quot;  An idiom for multi-tasking, not for failing to multi-task. Edit Addressing a few suggestions and another one that occurred to me:  &quot;Hedging your bets&quot; or &quot;Betting against yourself&quot;  In this case, you cannot win both bets, but you will almost certainly win one. You limit your upside, but also limit your risk. I'm looking in particular for total, catastrophic failure due to split attention/effort.  &quot;Can't walk and chew gum at the same time&quot;  Failing to do two fairly simple things that most everyone can do, not failing at two hard things because you tried to do both.  &quot;Robbing Peter to pay Paul&quot;  In this case, Paul gets paid. If you were to rob Peter to pay Paul, and get robbed yourself before you could get the money to Paul, that'd be closer. :)  &quot;Spreading yourself too thin&quot;  An actual English idiom, but missing the implied sense of failure.  &quot;Too many irons in the fire&quot;  &quot;A lot of irons in the fire&quot; can be viewed positively, so this is missing the sense of a strategic misstep.  &quot;Between a rock and a hard place&quot;  The sense is of needing to act but having no good alternative.  &quot;Falling between two stools&quot;  My favorite answer so far because it does appear to be an actual English idiom (not a proverb). In that sense, it's great, but some of the usages are closer to a failure of  than a failure of effort. And falling between two stools implies that the attempted action was simply sitting down.  &quot;Chasing two rabbits and catching none.&quot;  This is the exact sense I was looking for, but it appears to be a cross-cultural proverb rather than idiomatic English like &quot;putting all your eggs in one basket&quot;. I'll be accepting it, and using it in my work! But if there are other ideas, please let me know. PHP calculating number of days between 2 dates Omitting fields in Bibliography Who is blowing the horn to alert the Night's Watch? Setting git default flags on commands QGIS apply style to shapefile /dev/tcp not found When I try to run the following command:  echo -e "GET / HTTP/1.1\n\n" | /dev/tcp/74.125.225.19/80   I get the following error message:  bash: /dev/tcp/74.125.225.19/80: No such file or directory   The following command works perfectly, so the problem involves how I'm using /dev/tcp:  echo -e "GET / HTTP/1.1\n\n" | nc 74.125.225.19 80   I'm in Ubuntu 13.04, so the capability should be on my system. What am I doing wrong? What are the rules for using /dev/tcp properly?