Be $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, be $V$ and euclidean or unitary $\mathbb{K}$-vector space and be $v_1,\ldots,v_n \in V$.  For the $n\times n$ matrix  $$A:=(\langle v_j,v_k\rangle)_{j,k=1,\ldots,n}$$  then applys $A=A^*$ and $A\succeq0$.  Then $A$ is positive definite if and only if the vectors $v_1,\ldots,v_n$ are lineary independent.  I just can't seem to find a proof for this, and can't really proof it myself. Thanks in advance for any kind of help.  edit: the linked question doesn't talk about positive definity, or does it? Sorry if it does.
Given a set of vectors $v_1 \cdots v_n$, the $n\times n$ Gram matrix $G$ is defined as $G_{i,j}=v_i \cdot v_j$  Due to symmetry in the dot product, $G$ is Hermitian.  I'm trying to remember why $|G|=0$ iff the set of vectors are not linearly independent.
Command "find -name" must be enclosed in quotes or it doesn't work. Why is that? find . -name *.rb   doesn't work on my ubuntu  I have to do  find . -name "*.rb"   to get it to work.  Why is that? Replace $ with £ using cat Command I have file test_output_archive.svd with some Dollar signs ($). I have generated a shell script Replace_dollar_pound.sh with cat command to replace dollar symbol with Pound (£) script is working on Solaris 10, Dollar getting replaced as expected. In Solaris 11, but the dollar is not getting replaced, not throwing any errors  Content of test_output_archive.svd : Line Rental $9  OOB Only for $10  Replace Dollar symbol with Pound at step1, Remove junk charactes at step 2, Remove temp file at step3, Rename at step4**  Content of Replace_dollar_pound.sh : cat test_output_archive.svd | tr "\044" "\243" > temp_archive.svd cat temp_archive.svd | tr -d "\302" > test_output_archive2.svd rm temp_archive.svd mv test_output_archive2.svd test_output_archive.svd  Any Modifications to be done as OS is updated to Solaris 11 ?? Is there a way to disable the hotkeys? Is there a tool that combines zcat and cat transparently? When handling log files, some end up as gzipped files thanks to logrotate and others not. So when you try something like this:  $ zcat *   you end up with a command line like zcat xyz.log xyz.log.1 xyz.log.2.gz xyz.log.3.gz and then with:  gzip: xyz.log: not in gzip format   Is there a tool that will take the magic bytes, similar to how file works, and use zcat or cat depending on the outcome so that I can pipe the output to grep for example?  NB: I know I can script it, but I am asking whether there is a tool out there already. Cardinality of a totally ordered union Let $(X_{\alpha})_{\alpha \in A}, (Y_{\alpha})_{\alpha \in A}$ be families of sets such that for $\{X_{\alpha} \mid \alpha \in A \}$ and $\{Y_{\alpha} \mid \alpha \in A \}$ are totally ordered by inclusion. We have $(X_{\alpha},Y_{\alpha}) \neq (X_{\beta},Y_{\beta})$ for $\alpha \neq \beta$. Suppose that $|X_{\alpha}| \leq |Y_{\alpha}|$ for any $\alpha \in A$. It is true that $|\bigcup_{\alpha} X_{\alpha}| \leq |\bigcup_{\alpha} Y_{\alpha}|$? Nonlinear function continuous but not bounded I would like an example of a map $f:H\rightarrow R$, where $H$ is a (infinite dimensional) Hilbert space, and $R$ is the real numbers, such that $f$ is continuous, but $f$ is not bounded on the close unit ball $\{ x\in H : \|x\| \leq 1\}$.  Actually, $H$ could be replaced by any Banach space (but not just a normed space-- that's too easy).  My motivation is that if $f$ is linear, this is impossible; but I have next to no intuition about non-linear functions.  Edit: Here's an example for $c_0$ which is even differentiable (disclaimer: I found it here: ).  Define $f:c_0\rightarrow F$ (where F is your field, real or complex) by $$ f(x) = \sum_{n=1}^\infty x_n^n \qquad (x=(x_n)). $$  You can estimate the sum by a geometric progression, so it does converge.  A bit of checking shows that f is Frechet differentible (so certainly continuous).  But $f(1,1,\cdots,1,0,\cdots)=n$ (if there are $n$ ones) so $f$ is not bounded on the closed unit ball.  What I don't immediately see is how to adapt this to $\ell^2$, say. I lost my .keystore file? All odd primes except $5$ divide a number made up of all $1$s Okay, so I have worked on this problem and even though I can see it is true I just don't know how to show it.  Here it goes.  Question:  Show that every odd prime except $5$ divides some number of the form $111 \ldots 11$ ($k$ digits, all ones).  Thank you my friends! How to prevent a background process from being stopped after closing SSH client in Linux I'm working on a Linux machine through SSH (Putty). I need to leave a process running during the night, so I thought I could do that by starting the process in background (with an ampersand at the end of the command) and redirecting stdout to a file.  To my surprise, that doesn't work. As soon as I close the Putty window, the process is stopped.  How can I prevent that from happening?? Order of a element on a Symmetric Group Endnotes do not be superscript and add a space X,Y,Z missing in F3 When I press F3 to see the debug screen in I use Minecraft 1.8.1, some information is missing.   Most notably, there are no X, Y, Z coordinates.     What happened to my debug overlay? Can I apply for and receive a US visa when residing in country other than my own? I'm currently residing in a country other than my own, and am planning a business trip to the USA for a week. Being a Bulgarian citizen, I do not qualify for the ESTA visa waiver program, unlike other EU countries.  The US embassy in Sofia lists a  which they inexplicably describe as "really quite easy". Unfortunately, some of the steps involved would be a major inconvenience for me, as I'm not residing in the country currently, particularly the interview and the visa/passport delivery.  I was hoping it could be possible to apply for a visa from the country of current residence rather than my own, but I wasn't able to find any information on the US embassy's webpages concerning this. How to get the day, year, hours, min Individually from date format "yyyy-MM-dd'T'HH:mm:ss.SSSZ"? CSOM: Identify "columns" created by user rather than built-in When iterating the fields of a list how do you know which columns are yours and which are built-in ?  I know a solution for "classic" SDK using SPField.SourceId  but I can't find one for CSOM.  I thought UserCreated was the way to go for but   So how do you tell apart theirs and mine ? Using a combination of Hidden CanBeDeleted FromBaseType ? Checking for a list of special internalNames ?          ClientContext context = new ClientContext(siteUrl);         context.Credentials = creds;         Web web = context.Web;         var list = web.Lists.GetByTitle(listName);          context.Load(list.Fields);         context.ExecuteQuery();          foreach (Field field in list.Fields)         {         bool thisColumnWasCreatedByUser =                        field .FromBaseType == false &amp;&amp;                     //this.InternalName != "Title" &amp;&amp; // builtin but I'd rather keep it                     field .InternalName != "_dlc_DocId" &amp;&amp; // probably safe                     field .InternalName != "_dlc_DocIdUrl" &amp;&amp;                     field .InternalName != "_dlc_DocIdPersistId" &amp;&amp;                     field .InternalName != "Combine" &amp;&amp; // sucks                     field .InternalName != "RepairDocument" // sucks too         }   EDIT  Mixing suggestions by Viraj Gorajia and Dikesh Gandhi, my current solution is  bool thisColumnWasCreatedByUser =  field .FromBaseType == false &amp;&amp; (!field .SchemaXml.Contains(" SourceID=\"http")) &amp;&amp; // remove "Title", "Combine", "RepairDocument"  (!field .EntityPropertyName.StartsWith("OData__")) // field .Sealed == true could be working too but I don't know if it could exclude valid results Storing output of command in shell variable What if residuals are normally distributed, but y is not? I've got a weird question. Assume that you have a small sample where the dependent variable that you're going to analyze with a simple linear model is highly left skewed. Thus you assume that $u$ is not normally distributed, because this would result in normally distributed $y$. But when you compute the QQ-Normal plot there is evidence, that the residuals are normally distributed. Thus anyone can assume that the error term is normally distributed, although $y$ is not. So what does it mean, when the error term seems to be normally distributed, but $y$ does not? What is the meaning of the color of the node sockets in the node editor? I have been searching on the web and I can't find this. What is the meaning of the color of the node sockets in the node editor? Suppressing addition of blank page before figure when wrapping long JPEG image "Authoring" versus "Writing" In my area of work the word author as a verb has become quite common. However, it seems to have a subtly different meaning than plain vanilla write. For example, one does not write software unit tests; one authors them.  Can anyone offer a clear distinction between the two words? Has author as a verb become a common usage outside of the field of computer software?