So a matrix, A, is linearly independent if Ax = 0 has only the trivial solution (x=0) so let A = AB, then if the columns of AB are linearly independent, the equation (AB)x = 0 has only the trivial solution by the associative law of multiplication, (AB)C = A(BC), so (AB)x = A(Bx) = 0 I'm not sure where to go from here, or even if this is a proper way to go about proving that the columns of B are linearly independent
If you have an $m \times n$ matrix and the rows are linearly independent, are the columns linearly independent too?  I know that it's the case for $n \times n$ matrices. I've yet tried to think about it, and I think the answer is yes, for example the matrix: $\begin{bmatrix} 1 &amp; 2 &amp; 3 \\ a &amp; 2a &amp; 3c \end{bmatrix}$ has linearly dependent rows and columns, but I'm not sure that it works for every $m \times n$ matrix.
What is the best solution for database connection pooling in python? I have developed some custom DAO-like classes to meet some very specialized requirements for my project that is a server-side process that does not run inside any kind of framework.    The solution works great except that every time a new request is made, I open a new connection via MySQLdb.connect.    What is the best "drop in" solution to switch this over to using connection pooling in python?  I am imagining something like the commons DBCP solution for Java.    The process is long running and has many threads that need to make requests, but not all at the same time... specifically they do quite a lot of work before brief bursts of writing out a chunk of their results.  Edited to add: After some more searching I found  which looks decent, but as I'm relatively new to python I guess I just want to make sure I'm not missing a more obvious/more idiomatic/better solution. Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$ I've encountered the following problem in complex analysis:  Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$.  I've decomposed $f$ to it's real and imaginary parts and got that it's enough to prove that:  $(\int_0^{2\pi}cos(\theta)g(\theta)d\theta)^2 +(\int_0^{2\pi}sin(\theta)g(\theta)d\theta)^2 \le 16$   Where $g(\theta)=f(e^{i\theta})$.   But now I'm stuck. Any ideas?  Thanks For example, when I type ping 8.8.8.8 &amp;, why can we see ping process working? And when I type find / -name '*test*' &amp; , it's also seen on monitor.  Why is that? Is it not really in background? If I've got a file (or files) selected in Windows Explorer, the only way I've found to "unselect" them is click with the mouse on some empty space.  I thought Esc would do this, but it doesn't. Also changing focus with Tab or Shift + Tab doesn't work; the file(s) are still selected even if focus goes to the address bar for instance.  Does anyone know of a keyboard shortcut to do this? A question on limsup My Linux Mint 15 install on VirtualBox was working great up until about two days ago, when the updater started throwing a ton of errors and refuses to progress with any apt-get update, apt-get install, or apt-get upgrade.  They are 404 errors on the archive.ubuntu.com site, under the /ubuntu/dists/raring-updates folder. I looked at it, it doesn't exist.  a direct link, which shows there is no raring-updates.  Where has raring-updates gone and what can I do to fix this? Why won't vertical-align: middle work?  And yet, vertical-align: top does work.   span{   vertical-align: middle; } &lt;div&gt;   &lt;img src="https://via.placeholder.com/30" alt="small img" /&gt;   &lt;span&gt;Doesn't work.&lt;/span&gt; &lt;/div&gt; What's the bad magic number error? What's the "Bad magic number" ImportError in python, and how do I fix it?  The only thing I can find online suggests this is caused by compiling a .py -> .pyc file and then trying to use it with the wrong version of python. In my case, however, the file seems to import fine some times but not others, and I'm not sure why.  The information python's providing in the traceback isn't particularly helpful (which is why I was asking here...), but here it is in case it helps:  Traceback (most recent call last):   File "run.py", line 7, in &lt;module&gt;     from Normalization import Normalizer What issues / pitfalls must be considered when overriding equals and hashCode? Asking feedback/rating for a mobile app? Fig.1: A typical prompt, asking user for feedback/rating in iOS Background Any serious smartphone users must have seen a message similar to Fig.1 above at least once. Variations of such prompt include: &quot;Please give us 5 star ratings if you love this app!&quot;, &quot;Rate us :)&quot;, &quot;Your rating helps improve this app!&quot;, etc. Basically, Fig.1 asks user to leave a (nice) feedback/rating for the app on its AppStore(or equivalents).  The Shorter Question  When and how often should a mobile app user be asked to leave a feedback/ratings?   The Longer Question #1. Which User-statistics? What are the significant user's behavioral statistics that should be based on to generate Fig.1? For example:  Number of runs since install/reinstall of the application Time elapsed since current application run Time elapsed since first application run Time elapsed since first application install Certain events (e.g. after certain number of stages are cleared for a game app)  #2. When to Ask? If multiple user-statistics are chosen to be calculated to generate Fig.1, specifically how should the user-statistics satisfy the condition of the event? For example:  Form a unified equation (e.g. X^2+Y^2=Z^2) with all significant user-statistics partaken fairly to trigger the event. Create (multiple of) &quot;absolute&quot; event triggers (e.g. every 10th app run; elapsing 2 hours since current application run). Both of the two. Generate Fig.1 whichever comes true first. Neither of the two. &quot;There's a better way!&quot;  #3. How often? What would be the optimal min/max frequency of Fig.1 generation? By optimal, I mean users are not to be bothered too much, yet developers gain maximum output of great feedbacks from users? #4. &quot;Don't bother me again :(&quot; Should users have an option to opt-out from receiving Fig.1 at all? #5. Universal User-statistics for All Mobile Apps? Should apps under different categories be based on different user-statistics? There are significant differences in size and type of user-statistics in apps under different categories. For example, a &quot;Flashlight&quot; app is not typically run for one hour straight, while a music/movie streaming app may be run for over one hour. #6. &quot;This app does not need more feedback.&quot; Should every app of all kinds of user base ask this message? If any, should any mobile app stop generating Fig.1 when target conditions are met (e.g. downloads count, active user count, feedback count)? #7. Feedback/Rating Before Deletion Should users be asked to leave a feedback before deleting as seen in Fig.2 below? Users are more likely to leave very critical review on app deletion, but this kind of feedback will be very helpful for the developers to strengthen their app's weakness for next version release.                                                                                                 Fig.2: Asking user for rating before app deletion Normality of subgroups and Sylow groups under homomorphisms... Prove or disprove: if $f:G \rightarrow H$ is a surjective group homomorphism, then   1) $f(A)$ normal implies $A$ normal.  2) $S$ is a p-Sylow subgroup of $G$ implies $f(S)$ is a Sylow-p subgroup of $H$.   I believe 1 is false since I believe we need injectivity, but I cannot think of a counterexample...  I believe 2 is true but do not know how to show it... mod Distributive Law, factoring $\!\!\bmod\!\!:$ $\ ab\bmod ac = a(b\bmod c)$ I stumbled across this problem  Find $\,10^{\large 5^{102}}$ modulo $35$, i.e. the remainder left after it is divided by $35$  Beginning, we try to find a simplification for $10$ to get: $$10 \equiv 3 \text{ mod } 7\\ 10^2 \equiv 2 \text{ mod } 7 \\ 10^3 \equiv 6 \text{ mod } 7$$ As these problems are meant to be done without a calculator, calculating this further is cumbersome. The solution, however, states that since $35 = 5 \cdot 7$, then we only need to find $10^{5^{102}} \text{ mod } 7$. I can see (not immediately) the logic behind this. Basically, since $10^k$ is always divisible by $5$ for any sensical $k$, then: $$10^k - r = 5(7)k$$ But then it's not immediately obvious how/why the fact that $5$ divides $10^k$ helps in this case. My question is, is in general, if we have some mod system with $a^k \equiv r \text{ mod } m$ where $m$ can be decomposed into a product of numbers $a \times b \times c \ \times ...$, we only need to find the mod of those numbers where $a, b, c.....$ doesn't divides $a$? (And if this is the case why?) If this is not the case, then why/how is the solution justified in this specific instance? Background image resize in Ubuntu 18.04.1 LTS Recently updated to 18.04.1 and now I cannot resize my background image from fill screen to center image.  How can I resize the image now that 18 doesnt have the quick resize feature? Set the size of pgf picture? I have a pgf picture which I insert with:  \input{m1.pgf}   But the image is too big. Instead of regenerating a smaller image, I'd rather take advantage of vector graphics, and have XeLaTeX scale it. How do I set the width of pgf picture (keeping its aspect ratio)?  MWE  tex:  \documentclass{article}  \usepackage{fontspec} % enagles loading of OpenType fonts \usepackage{polyglossia} % support for languages  % fonts: \defaultfontfeatures{Scale=MatchLowercase,Mapping=tex-text} % without this XeLaTeX won't turn "--" into dashes \setmainfont{DejaVu Sans} \setsansfont{DejaVu Sans} \setmonofont{DejaVu Sans Mono}  % pgf: \usepackage{pgf}  \begin{document}  \input{m1.pgf}  \end{document}   Sample pgf:  (it's too big to cite it here, and I can't make much smaller). Installed Teamviewer using a 64-bit system, but I get a dependency error Why did the Emperor remain motionless while Darth Vader threw him off the balcony? I'm wondering whether this was a movie sin but his body position seemed to stay the same? Also why didn't he stop using Force Lightning? He continued to use it even as he was falling down.   I'm also wondering whether Palpatine meant to electrocute Vader or was it just a coincidence (because he kept firing Force Lightning) .  It makes no sense as he just stood there allowing Vader to throw him. Grammatical number agreement in a complex phrase using singular "they" Why split the  tag when writing it with document.write()? Why do some sites (or advertisers that give clients javascript code) employ a technique of splitting the &lt;script&gt; and/or &lt;/script&gt; tags up within document.write() calls?  I noticed that Amazon does this as well, for example:  &lt;script type='text/javascript'&gt;   if (typeof window['jQuery'] == 'undefined') document.write('&lt;scr'+'ipt type="text/javascript" src="http://z-ecx.images-amazon.com/images/G/01/javascripts/lib/jquery/jquery-1.2.6.pack._V265113567_.js"&gt;&lt;/sc'+'ript&gt;'); &lt;/script&gt; There's something funky about some titles in revision histories here on Meta Stack Exchange Well, this might be a really simple one. But still... What will be the soln. to --- \begin{aligned} \int\frac{1}{1+x^n} dx \end{aligned}  Is substituting \begin{aligned} 1+x^n \end{aligned} by tan z the correct step?  Thanks for the Help.