Can anyone help me understand why is this true?  I took a couple of examples and it somehow sorts out to be true, but I don't know how to somehow prove it and generalize.  any help is appreciated :)
Prove that $Ax=0$ has infinite solutions if and only if $ \text{rank}(A)&lt;n$   Here is one way of my if proof, but I don't know how to proof the other part.  Let $r$ be the rank of $A$. Then $ r\leq  n$. If $ r&lt;n$, then there are $n-r$ linearly independent solutions. Furthermore any linear combination of these solutions will also be a solution of $Ax=0$. Hence, in this case, the equation $Ax=0$ has infinite number of solutions
Last Logon Time of All Users - RDP server not on a domain Can I change some but not all Apple keyboard function keys (dim vs F1, bright vs F2)? I want to override F1 and F2 behavior, since I have external display and Mac can't control display's brightness. The rest of the keys should retain their non Fx  I know about "Use all F1-F12 keys as standard function keys.", but I don't want to change all of the keys to be F when pressed, just some of them. (For me single press music and volume control is necessary.)  Is there a way to pick some of the function keys to default to numbered function instead of the "Apple" definitions system wide? How come such different methods result in the same number, $e$? This works just fine:      protected void txtTest_Load(object sender, EventArgs e)     {         if (sender is TextBox) {...}      }   Is there a way to check if sender is NOT a TextBox, some kind of an equivalent of != for "is"?  Please, don't suggest moving the logic to ELSE{} :) Suppose I have something like the following:      \begin{eqnarray*} \mathrm{gcd}(x^3 + 6x^2 + 5x + 5, x^3 + 13x^2 + 6x + 3) &amp;=&amp; 102x^2 + 108x + 2 \ \mathrm{with} \ p = 109 \\ \mathrm{gcd}(x^3 + 2x^2 + 9x + 4, x^3 + 3x^2 + 7x + 9) &amp;=&amp; 1 \ \mathrm{with} \ p = 131 \\ \mathrm{gcd}(x^3 + 3x^2 + 9x + 12, x^3 + 6x^2 + 12x + 4) &amp;=&amp; 6x + 122 \ \mathrm{with} \ p = 157 \end{eqnarray*}   How do I align the statements at the end of each line (i.e. the "with p = ..." bit) to the right? Calculating MEAN and STANDARD DEVIATION confussed [] ** calculate the mean and standard deviation of each investment, assume that both returns come from independent samples, are both normally distributed and that the (true but unknown) variances are the same,, Can someone could check my  calculations for  mean and standard deviation? I did calculations but not sure what to do with negative values of distances(If if am correct Investment A MEAN= 78.6875 when it comes to calculating standard deviation i have to power difference by 2 ?  ** A 256GB SSD has Windows-10 and 16.04 installed.  BIOS not UEFI.  A charity I work with requires me to use Quickbooks Desktop on Windows.  The plan is to ghost the existing system to a 512GB or larger SSD and create a third bootable Windows 7  partition for performing charity activities.   Is this possible to have a third partition? Is there a good example you have used as a guide? Is triple booting with two Windows partitions different from 2 Linux partitions?   If you have done this before ( with GRUB) please indicate this in your response: lessons-learned are appreciated.  Thank you What is the difference between “say” and “tell”? I really get confused when to use say and when to use tell. Which is appropriate in the following:   What did he tell? / What did he say? What are you saying? / What are you telling? Getting filesystem path of class being executed How can I pad an integer with zeros on the left? When I log in with LightDM on my laptop running Debian Unstable, it recently started to hang for around 2 minutes until journalctl shows the message kernel: random: crng init done. When I press random keys on my keyboard while it hangs, it logs in faster (around 10 seconds). Before I didn't have this issue, is there any way I can fix it?  Edit: using linux-image-4.15.0-3-amd64 instead of linux-image-4.16.0-1-amd64 works, but I don't want to use an older kernel. difference between "function foo() {}" and "foo() {}" I can define bash functions using or omitting the function keyword. Is there any difference?  #!/bin/bash  function foo() {   echo "foo" }  bar() {   echo "bar" }  foo  bar   Both calls to functions foo and bar succeed and I can't see any difference. So I am wondering if it is just to improve readability, or there is something that I am missing...  BTW in other shells like dash (/bin/sh is symlinked to dash in debian/ubuntu) it fails when using the function keyword. How to get rid of blank pages in PDF exported from SSRS I have a two-page SSRS report. When I exported it to PDF it was taking 4 pages due to its width, where the 2nd and 4th pages were displaying one of my fields from the table. I tried to set the layout size in report properties as width=18in and height =8.5in.  It gave me the whole table in a single page of PDF, but I am still getting the 2nd and 4th pages blank.   Is the way I am doing it incorrect?  How else can I get rid of those blank pages? Created user can access all databases in PostgreSQL without any grants Zip all files in directory? Is there a way to zip all files in a given directory with the zip command? I've heard of using *.*, but I want it to work for extensionless files, too. Is there an Extended Backus–Naur form (EBNF) that covers all of English? surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism?  I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to preserve the property of being immersion.  I also noticed that if we replace $\mathbb{R}^2$ with $S^2$ or any other compact simply connected manifold, no such immersion exists, because then it would be a covering map (as a local diffeomorphism from compact space, thus proper), but since the manifold is simply connected, it must be a diffeomorphism. Python mechanize javascript Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial. My question is in the title. I'm a little lost in how to solve this problem. There is a hint associated with the problem that states the following:   Use a line integral to show that $ |f(z)| \leq A + |z| $ where A = $\max\{1, |f(0)|\}$  Thanks for the help. How do I get resolvconf to regenerate resolv.conf after I change /etc/network/interfaces? After updating /etc/network/interfaces with something very similar to below, how do I get /etc/resolv.conf to update?  I tried (as root) resolver -u, service networking restart but they didn't work. I also fixed the symlink for resolv.conf and tried resolver -u again. Finally in frustration I rebooted, which did fix the problem by rebuilding /etc/resolv.conf.  Ubuntu 12.04, 64bit server, all the latest patches installed.  Example /etc/network/interfaces:  iface eth0 inet static     address 192.168.3.3     netmask 255.255.255.0     gateway 192.168.3.1     dns-search example.com     dns-nameservers 192.168.3.45 192.168.8.10