Let $A$ be a $3\times 4$ and $b$ be $3\times 1$ matrix with integers entries. Suppose that the system $AX=b$ has a complex solution. Prove that the system $AX=b$  has a rational solution and set of all real solution of $AX=0$ has a basis consisting of rationals solutions ?.
Let $A$ be a $3\times 4$ and $b$ be a $3\times 1$ matrix with integer entries.Suppose that the system $Ax=b$ has a complex solution. Then which of the following are true? (CSIR December 2014)         $Ax=b$ has an integer solution.    $Ax=b$ has a rational solution.   The set of real solutions to $Ax=0$ has a basis consisting of rational solutions.   If $b$ is not equal to zero then $A$ has positive rank.      Is it possible to say that 1 and 2 are true as $Ax=b$ has a complex solution? If not  what we have to understand from the statement "the system $Ax=b$ has a complex solution " ? What about 3 and 4  ?
In my D&amp;D 5e campaign one of the PCs who's a wood elf started dating a dark elf, and as a result she got pregnant and she is about to give birth. One of the other players had their PC die and he is considering playing as the child of the couple.  My question is, which race is the child going to be? Is it okay to use Stack Overflow as the support forum for a product or project? Does the lack of sleep affect my health? In the past month, I've been really busy with the university and stuff, and I don't sleep as much as I used to. I usually sleep 4-5 hours a night.  I haven't been doing that extensively ( I always kept a good sleeping routine), but can the lack of sleep affect my health on the long term ? Managing multiple asynchronous NSURLConnection connections Irreducibility check for polynomials not satisfying Eisenstein Criterion. Why was C-3PO's memory wiped, but not R2-D2's? In Revenge of the Sith after Luke and Leia are born, they are sent to separate parts of the galaxy. The droids are given to Captain Antilles, with the instructions to clean them up, and have C-3PO's memory wiped. Why just C-3PO's and not also R2-D2's?   Initially I thought it was due to R2-D2's inability to communicate verbally, but he not only can communicate via text message (as seen in The Empire Strikes Back I believe), but we see other characters understanding his blips and bleeps as well. Here's an example of a Google Docs Spreadsheet script that sends emails using Google spreadsheets. I want the email Sent in HTML not plain text. Can someone help me out?  function sendEmails() {   var sheet = SpreadsheetApp.getActiveSheet();   var startRow = 2;  // First row of data to process   var numRows = 2;   // Number of rows to process   // Fetch the range of cells A2:B3   var dataRange = sheet.getRange(startRow, 1, numRows, 2)   // Fetch values for each row in the Range.   var data = dataRange.getValues();   for (i in data) {     var row = data[i];     var emailAddress = row[0];  // First column     var message = row[1];       // Second column     var subject = "Sending emails from a Spreadsheet";     MailApp.sendEmail(emailAddress, subject, message);   } } Installing Autodesk Maya on Ubuntu is possible, but can be a bit challenging.  Assuming that one already has the required specs and a solid graphics card driver installed, how can it be done? Why do only two sexes exist for animals? Why, from the natural selection point of view, do only two sexes exist for animals? I want to enable telnet on Windows 10.  I only see a telnet client but not a telnet server.  There is a web page telling me to open server manager and do a lot of stuff.  But, I don't have a server manager.  Please help!  I want to enable telnet. Which registry keys are impacted with Xerox Printer driver installation? I am implementing a driver deployment with Samba.  Every possible model of printer found an easy solution, except for Xerox Workcenter.    After browsing the Internet, I found that Xerox printer driver has to change some registry keys in order to work, keys which are changed by the executable downloadable on the manufacturer website.  Do you know which key are impacted by the Xerox Workcenter GPD driver?  This corresponds to the error message I get while uploading to Samba, which asks the position of a file named as a registry key.  I browse Samba bug tracker and they did not specify which registry key are impacted.  I contacted Xerox itself, but despite that fact that we have a support contract and really a huge number of workstations, it seems that Xerox is not willing to help us. $n$ divides a product of $n$ consecutive integers (by $n$ divides one of them) How do I get a thread ID from an arbitrary pthread_t? I have a pthread_t, and I'd like to change its CPU affinity. The problem is that I'm using glibc 2.3.2, which doesn't have . That's OK, though, because pthread_setaffinity_np() is itself a wrapper of , which can be called by passing a thread ID instead of a process ID to set the affinity for an arbitrary thread.  BUT ... The thread id that sched_setaffinity can work with is an OS thread id, the kind that you can get from the  system call. This is different from the opaque type pthread_t, and gettid() will only return the thread-id of the current thread. I need to be able to set the CPU affinity of an arbitrary thread.  Unfortunately, I can't access the pthread's private parts, which would let me steal the thread id by casting a pthread_t to a struct pthread *. All the better, I guess, since relying on private implementations is asking for even more trouble.  I've also been reading up on the  function, however this returns a "unique integral identifier" -- which I don't believe is in any way shape or form equivalent to an OS thread id.  Hence, the question: How can I get a thread ID from an arbitrary pthread_t? I have no idea how to find this integral:  $$\int_0^1\frac{(x-x^2)^4}{1+x^2}\ dx\ ?$$  Any help would be appreciated. Thanks in advance. A better answer was posted long after accepting another. Should I accept the new one? Where is the .Trash folder? I would like the user to have full rights on this folder (as well as all sub-directories and files in it):  ~/.blabla   currently owned by root.  I have found numerous posts (in this forum and elsewhere) on how to do this for files but I can't find a way to do it for whole folders. Let $r$ be a relation on $A \times A$ such that $(a,b) r (c,d) \iff ad = bc.$ How can I show that this relation is transitive, ie. $(a, b)r(c,d)$ and $(c,d)r(e, f) \implies (a,b)r(e,f)$?  I tried to say that $(a,b) r (c,d)$ means that $c=ka$ and $d=kb,$ $k$ a coefficient, such that $ad = bc \iff akb = bka,$ and going on from there, but I'm not sure this is valid for all values of $a, b, c,$ and $d$. How can I show the transitivity of the relation? What is the best way to flatten a linked list with lists as values? I'm trying to determine if there is a best practice for typesetting the "evaluated at" bar for derivatives. The three possibilities I'm thinking of are shown in the code excerpt below:  \begin{gather*} 5 + \frac{df}{dt} \Bigr\rvert_{t = 0} \\ 5 + \frac{df}{dt} \biggr\rvert_{t = 0} \\ 5 + \left. \frac{df}{dt} \right\rvert_{t = 0} \\ \end{gather*}     The \Bigr option doesn't quite reach the top and bottom of the fraction. The \biggr option overshoots the top and bottom of the fraction by a little bit. And the \left...\right option has the same height as the \biggr option, but it introduces a little extra space between the + and the fraction. I'm having trouble deciding between these options. Is there a best practice for typesetting this?