Suppose matrix A has rank r. Pick any r independent rows and r independent columns from A. Must the submatrix formed from these rows and columns be invertible?
Let $A$ be an $m \times n$ matrix of rank $r$, let $I$ be a set of row indices such that the corresponding rows of $A$ are independent and let $J$ be a set of $r$ column indices such that the corresponding columns of $A$ are independent.  Let $M$ denote the $r \times r$ submatrix obtained by taking rows from $I$ and columns from $J$.  Then $M$ is invertible.  So far I've got: Let $B$ be the $m \times r$ matrix obtain from taking the rows from $J$.  $B$ is row-reducible to having a pivot in each column or else $Ax = 0$ has more than one solution, a contradiction.  Also, the dimension of the row space of $B$ equals $r$ as well.
Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. Didn't understand relationship 'AccountContactRelation' in FROM part of query call I'm getting the following error message:     System.QueryException: Didn't understand relationship 'AccountContactRelation' in FROM part of query call. If you are attempting to use a custom relationship, be sure to append the '__r' after the custom relationship name. Please reference your WSDL or the describe call for the appropriate names.   Select Email, Id, (Select ContactId, AccountId From AccountContactRelation) From Contact  In AccountContactRelation object there are two lookup fields for Account and for Contact.  Please advise how to avoid such error message? Using SQL Server, how do I split a string so I can access item x?  Take a string "Hello John Smith". How can I split the string by space and access the item at index 1 which should return "John"? Diagonalisation and characteristic polynomial Let $A\in M_3(\Bbb R)$ which is not a diagonal matrix. Let $P$ be a polynomial (in one variable), with real coefficients and of degree 3 such that $P(A) = 0$. Pick out the true statements:  a. $P = cP_A$ where $c \in\Bbb R$ and $P_A$ is the characteristic polynomial of $A$;  b. if $P$ has a complex root (i.e. a root with non-zero imaginary part), then $P = cP_A$, with $c$ and $P_A$ as above;  c. if $P$ has a complex root, then $A$ is diagonalizable over $\Bbb C$.  I guess b and c are true although I am unable to disprove a. Because A could satisfy $(X-A)$ but this is not a polynomial. Now if $P$ has a complex root, then its conjugate too is its root as the polynomial has  real coefficients. As we all know An odd degree polynomial has always a real root, what we have is three distinct roots. Since $A$ satisfies the polynomial the only difference between the characteristic polynomial and $P$ has to be a constant. If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$. Let $a,b,c$ be the lengths of the sides of a triangle. Prove that $$\sum_{\text{cyc}}\frac{a}{b+c}=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}&lt;2\,.$$  Attempt.  By clearing the denominators, the required inequality is equivalent to $$a^2(b+c)+b^2(c+a)+c^2(a+b)&gt;a^3+b^3+c^3\,.$$ Since $b+c&gt;a$, $c+a&gt;b$, and $a+b&gt;c$, the inequality above is true.  Is there a better, non-bruteforce way? How to start a GUI software on a remote Linux PC via SSH Sometimes I need to start XMBC media player or other GUI software a one of my PC (small Xubuntu PC used as a media center).  Usually I do this starting a X11vnc server on the remote PC via SSH and than connecting with an Xvnc client to the Xfce desktop.  Is there a way to start a GUI software on a remote Linux PC via SSH?  Thanks! Find the smallest disk in complex plane containing the eigenvalues of a matrix (NBHM 2012) Consider the given matrix  $$\left[\begin{matrix}1&amp; -2&amp; 3&amp; -2 \\1&amp; 1&amp; 0&amp; 3\\-1&amp; 1&amp; 1&amp; -1\\0&amp; -3&amp; 1&amp; 1&amp;\end{matrix}\right]$$  Find out the smallest disk like ($|z-1| &lt; r$ ) in the complex plane containing the eigenvalues of the given matrix.  One easiest method is to find out the eigenvalues from the characteristic equation of the matrix. I can do it so no problem with it.   I am looking for a method to solve the problem without the above method that can be evaluated in very short time. This type of method is possible, I believe.  Thank you for your help About an hour ago, from my account, an inappropriate message was posted as a reply to a Stack Overflow question.  I have noticed it, because I started getting downvotes. To my understanding my Gmail account wasn't compromised (I reviewed my account security settings and login history). I am from a secure WPA2 connection. I cannot find enough information on how to investigate this. Although I changed my passwords everywhere, I still seem confused and I would like to ask how to investigate this issue myself. "Similar questions" overlaps footer and has no background When I can scroll down far enough to have the hot network questions overlap the footer (because my question is short) there is no background under the text:    Could they either be given a white background: (just disable the transparency option)    Or stop at the footer:      This also happens with the help text on some sites, as per this bug report:     Ubuntu 14.04.1 (Latest LTS Version), . Same problem with  and .  Windows 10 Pro,  and .  (Images linked to screenshot of browser)  My question is different from  because my issue is the lack of background on the footer.   I think there was a site redesign? Apt "could not find a distribution template" error When I attempt to install anything using apt-get I get the following error:  Traceback (most recent call last):   File "/usr/bin/add-apt-repository", line 60, in &lt;module&gt;     sp = SoftwareProperties()      File "/usr/lib/python2.6/dist-packages/softwareproperties/SoftwareProperties.py", line 90, in __init__     self.reload_sourceslist()   File "/usr/lib/python2.6/dist-packages/softwareproperties/SoftwareProperties.py", line 538, in reload_sourceslist     self.distro.get_sources(self.sourceslist)       File "/usr/lib/python2.6/dist-packages/aptsources/distro.py", line 90, in get_sources     raise NoDistroTemplateException("Error: could not find a " aptsources.distro.NoDistroTemplateException: Error: could not find a distribution template   Any idea what this means and how to resolve it? primitive n-th roots of unity Show that the primitive n-th roots of unity have the form $e^{2ki\pi/n}$ for $k,n$ coprime for $0\leq k\leq n$.  Since all primitive n-th roots of unity are n-th roots of unity by definition they all have that form, the question is, how to show $k$ and $n$ are coprime. How can I automatically shutdown the system after a specific time? L293, L298 and SN754410 H-bridge drivers on low voltage power supply This question is about the following three integrated H-bridge drivers: -    or L293D (D = protection diodes added)  (protection diodes included)  (no protection diodes)   Time after time the same question keeps coming up - someone is using one of these devices (on a low voltage, usually around 6V or less) and they are just not performing adequately. The reasons are listed further below but my question is this: -  What H-bridge drivers are preferred when controlling a low-voltage motor?   Information  The L293 and the SN754410 are nearly identical and crucially, if you try and control a 1 amp load, you are faced with dismal performance: -    The tables tell you (typical conditions) that the upper transistor drops (loses) about 1.4 volts when driving a 1A load and, the lower transistor drops (loses) about 1.2 volts when driving a 1A load. The upshot is that if you have a 6V, 1A motor and 6V battery, don't expect to see more than 3.4 volts across the motor: -  \$V_{OUT} = 6V - (1.4V + 1.2V) = 3.4V\$  Worst case scenario is you might only see 2.4 volts across it.  What about the L298? It's got a nice big heat-sink whereas the L293 and SN754410 are regular-looking chips. Here's what the volt drop (losses) look like: -    It's the same story - for a 1A load, you can expect to lose up to 3.2 volts and, what you thought might be 6V across your motor, is at best 4.2 volts and at worst only 2.8 volts.  Clearly none of the devices listed are suitable for low voltage applications where the motor might be expected to draw in excess of 0.5 amps. Let's say I'm running a simple server and have accept()ed a connection from a client.  What is the best way to tell when the client has disconnected? Normally, a client is supposed to send a close command, but what if it disconnects manually or loses network connection altogether? How can the server detect or handle this? $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$ I've been doing some more exercises in , in particular the following:  Show that $H_0(X,A) = 0$ iff $A$ meets each path-component of $X$.  "$\Leftarrow$": Let $x_i \in A \cap X_i \neq \emptyset \forall $ path-components $X_i$. Then for any $x \in X_i$ there is a path $\gamma$ from $x_i$ to $x$. They therefore differ by a boundary: $\partial \gamma = x - x_i$ and therefore $\{&#160;x \} = \{ x_i \}$ in $H_0(X,A) $. Therefore $H_0(X,A) = 0$.  "$\Rightarrow$" Now for this direction I'm not so sure and I'd be glad if you could give me a hint. Let $H_0(X,A) = 0$. By definition, the relative homology group is calculated from the following sequence of chain groups:  $$ 0 \rightarrow C_0(A) \rightarrow C_0(X) \xrightarrow{\partial_1} C_0(X) / C_0(A) \xrightarrow{\partial_0} 0$$  Then $H_0(X, A) = H_0(C_0(X) / C_0(A)) = ker \partial_0 / im \partial_0 = (C_0(X) / C_0(A) ) / im \partial_1 = 0$.  There are two cases where this equality holds:  (i) $C_0(X) / C_0(A) = 0$  (ii) $im \partial_1 = C_0(X) / C_0(A)$  In case (i), $A \cap X_i \neq \emptyset$ for all $X_i$.  In case (ii) I'm not sure how to proceed. Can you give me a hint? Many thanks for your help!  Edit Or maybe I could do the second direction like this:  $$0 = H_0(X,A) = \oplus_i H_0(X_i, A) \implies  H_0(X_i, A) = 0 \forall i$$  $$ \implies A \cap X_i \neq \emptyset \forall i$$? Get content from a url using php Let $\mathbf{FinVec}$ denote the category of finite-dimensional real vector spaces and linear maps, let $F\colon\mathbf{FinVec}\rightarrow\mathbf{FinVec}$ be a functor. For each pair of objects $V,W$, the Hom-set $\operatorname{Hom}(V,W)$ is itself a finite-dimensional vector space and we equip it with the unique structure of a topological space/of a smooth manifold that is compatible with its vector space structure. The composition maps are then continuous/smooth. In this way, we view $\mathbf{FinVec}$ as enriched over the category $\mathbf{Top}$/$\mathbf{Diff}$ respectively. It then makes sense to ask whether the functor $F$ is enriched over $\mathbf{Top}$/$\mathbf{Diff}$. That is just to ask whether the induced map $F\colon\operatorname{Hom}(V,W)\rightarrow\operatorname{Hom}(FV,FW)$ is continuous/smooth for all $V,W$. (Irrelevant to the rest of the question, but the reason this is a useful notion is that it is the natural hypothesis assuring that the functor induces another functor on the category of topological/smooth vector bundles, applying the original functor fiberwise.) In the case $V=W$, by functoriality, we can restrict $F$ to a group homomorphism $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$. If the functor is continuous/smooth, so is this map. Note that $\operatorname{GL}(V)$ is a Lie group. It is a general result that a measurable group homomorphism between Lie groups is automatically smooth. This begs the following questions:  What's an argument for the existence of functors that aren't continuous? Can we construct one? Given the above, it either happens that the restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$ are continuous and continuity fails elsewhere, or the induced maps will be non-measurable, so that the construction will necessarily have to involve choice to some extent. Are there continuous functors that aren't smooth? Given the above, a continuous functor will have smooth restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$, but smoothness may fail elsewhere. This question is inspired by my own answer to a  which I tried to answer and got stuck at one point.     The question was:     HI DARLING.      USE MY ATM CARD, TAKE ANY AMOUNT OUT, GO SHOPPING AND TAKE YOUR FRIENDS FOR LUNCH.      PIN CODE: $\displaystyle \int_{0}^{1} \frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 - 3x + 2}} \, dx $      I LOVE YOU HONEY.      Anyone knows? Are we gonna get an integer number?     My attempt:  Does this help?     $$\frac{3x^3-x^2+2x-4}{x-1}=3x^2+2x+4$$   (long division)   \begin{align*} I&amp;=\int\frac{3x^3-x^2+2x-4}{[(x-1)(x-2)]^{1/2}} dx = \\ &amp;=\int\frac{(3x^2+2x+4)(x-1)^{1/2}}{(x-2)^{1/2}} dx = \\ &amp;=\int 3(u^4-4u^2-4)(u^2+1)^{1/2}du \times 2 \end{align*}   after the substitution   \begin{gather*} (x-2)^{1/2}=u\\ du=\frac1{2(x-2)^{1/2}}dx\\ u^2=x-2\\ (x-1)^{1/2}=(u^2+1)^{1/2} \end{gather*}   Update:  may help us proceed.    I tried to proceed:     $$6\int (u^4-4u^2-4)(u^2+1)^{1/2} du = 6\int ((t-3)^2-8)t \frac{dt}{2u}$$   after $u^2+1=t$ and $dt=2udu$   \begin{align*} u^4-4u^2-4 &amp;= (u^2+1)^2-(6u^2+5) \\ &amp;= (u^2+1)^2-6(u^2+1)+1 \\ &amp;= ((u^2+1)-3)^2-8 \end{align*}   I wonder whether this question can be solved from here?    Update:   This has been getting a lot of views, and I think most people came for the sort of problem mentioned in the title (where I got stuck) rather than the original problem itself.  Keepin this in mind, I'm reopening the question and here's the kind of answers I expect — Solutions to the original problem are good, but I'd prefer solutions that continue from the part where I got stuck — the polynomial in $u$ — that's the sort of problem mentioned in the title. I've always been one to simply use:  List&lt;String&gt; names = new ArrayList&lt;&gt;();   I use the interface as the type name for portability, so that when I ask questions such as these I can rework my code.    When should  be used over  and vice-versa? Imported CSV points are not visible I am trying to import  CSV file and get no reported errors but there are no points visible. If I click on "zoom to extent" nothing happens (even if I move away from the location of the points before doing this.  I have shut off all raster layers but they were not hiding the points.  I have tried to find an answer here but none seem to fit my problem. I have followed the steps in this tutorial:   Here is an image of the CSV file:    here is the import dialog box:   and here is the QGIS attribute table after import:   I am saving in WGS84.  I tried the local projection (NAD83 Zone 19N) but that made no difference.   I am using QGIS 2.12.1 on Windows 10. Any suggestions on what I am doing wrong?