The question asks to use a proof by contradiction. I know that the coefficients of $\{v_1,v_2,v_3,v_4\}$ are all zero, but then I don't know what to do next-how to translate that into the $\Bbb R^3$ vector. Any help would be appreciated, thanks!
My problem: Prove that a set of vectors $S$ is linearly independent if and only if any finite subset of $S$ is linearly independent.  I tried like this:  Suppose S is LI.Then the vector $0$ cannot be expressed as a linear sum of all elements of $S$.  How it follows that a finite subset is also LI from this fact. I think $S$ can be finite or infinite.  This is a question from the book Linear Algebra - Friedberg et al.
Submitting a URL to Google using PHP/cUrl It just struck me that Utility Companies world-wide supply domestic users at either 50/60Hz. It could have been selected to be 10 Hz, or 100 Hz ... whatever  Was 50/60 chosen at random? Is there some rationale involved (efficiency, easier to construct/control etc) in restricting domestic supply to either 50/60 Hz?  In other words, if I were designing a power system, why would I pick 50/60 Hz instead of some other frequency? Custom function won't refresh as inputs are changed How to write a function (piecewise) with bracket outside? How to write this Piecewise function using LaTeX?  I tried $ \begin{array}{cc}   \{ &amp;      \begin{array}{cc}       0 &amp; x\leq 0 \\       \frac{100-x}{100} &amp; 0\leq x\leq 100 \\       0 &amp; 100\leq x     \end{array} \end{array} $  but it didn't work perfectly, the bracket &quot;{&quot; looks small, see: Implementing a Custom Subscription Center in Salesforce Marketing Cloud Has anyone implemented a custom subscription center using landing pages? The example one from the documentation doesn't seem to be working. I would really like to just customized the out of the box one but the amp script is hidden when I view source. Does anyone have and tips/tricks for creating a sub center?  Thanks! "have" vs."have got" in American and British English User suggested great corrective edit to my answer, but the edit was rejected. What can I do to approve it? Let's say I have a node type "Article" which has a field "File" which is set to handle public files. A lot of articles and files get added.  Now the specs suddenly change and the "Article" will move behind a login and therefore also the attached files must be changed to be not publicly accessible anymore.  I already found out  pretty easily by reimporting the manually updated field.storage.node.field_file.yml config. But: this only affects new files that will be added after the switch. It doesn't work for existing files. They still are accessible publicly. Many thousand existing files would need to be manually reuploaded to also make them private.  Does anybody of you have solved that in D8 before? If so, then how?  As all I can think of right now is to programmatically query all "Article" nodes, then foreach through all nodes, get the files, temporarily store them somewhere else, empty the field out, then try to reupload the files programmatically (probably my next question) and then resave each node.  Does that sound reasonable? Or can you think of any other approach that might work here (in D8)? How to add custom Modules in magento verifying an incorrect integrating factor I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is:  $-y^2dx +x^2dy = 0$  He then asks us to verify that:  $ u(x, y) = \frac{1}{(x-y)^2}$  is an integrating factor.   I multiplied through to get:  $\frac{-y^2}{(x-y)^2}dx + \frac{x^2}{(x-y)^2}dy = 0$   However, the partial derivatives of these do not equal each other so I am a bit confused... Is it possible to install an iPhone app in MacBook Pro? How do I install make? (I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)  How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above. How can I fix "unbalanced brackets" in my data tag? How to find the URLs accessed by iOS Apps? I am a third year (starting fourth year in the fall) PhD student in mathematics. I've passed all qualifying exams and am currently doing research. As far as I can tell, I am not doing poorly. I have the good fortune of having a great advisor, being in a very supportive department, and having friends and family who genuinely care about my success.   The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall. I am not complaining about the material being hard, and I am not averse to putting in hard work, but I get frequently discouraged when I realize the vast volumes of mathematics that I yet know nothing about (and probably never will). It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked. I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.   I've talked about this to some number of people; here is some advice I have received:   Take a day off. Putting in hours upon hours of trying things doesn't magically lead to a solution, particularly if the brain is tired and just wants to sleep. Have a hobby. Since math research doesn't exactly provide instant gratification, a hobby might provide some instead.  Talk to other graduate students. Realize that many graduate students go through this.   I'm interested to know how other folks have dealt with being discouraged as a graduate student. Does it get better with time and experience? Is this a sign that research is not for me and that I should seriously consider a life outside academia? Should reputation points expire? While looking at the SO question  It occurred to me that this was an extremely simple question with an extremely simple answer.  And yet, it has garnered the OP thousands in upvote points, simply because he was the first one to ask this question in the early days of SO.  There are many examples of these simple questions and I believe they don't accurately model a user's participation in the SO community.  People shouldn't be getting thousands or even tens of thousands of points from a question just because they were lucky enough to get it in the system a while ago.  Another related issue is that the non-expiring point system rewards users for seniority.  Simply put, the longer you're around SO, the higher your score will be.  While there's some merit to this, it seems to have a disproportionate effect.  So what can we do to limit these effects?  I'm just brainstorming but here are a couple ideas:   Automatically close voting on questions after a certain time period.  Good questions will still be rewarded, but they won't give disproportionately large rewards simply because they are really old questions. Expire users' points after a certain time period.  If you expire users' points after, say 3 months, then this will force them to stay active and their points will be more a reflection of "How much have you participated lately"?  It forces old users to stay active rather than resting on their laurels, and allows junior members a chance to catch up to senior members of the community by putting in a lot of effort.  Currently, there is no chance at all for junior members to ever catch up with senior members and their only 'sin' is that they didn't register 5 years ago. My Indian passport book cover is detached from the main book, but all other information, as well as the book is fine. I am traveling to India in 2 days. What should I do? Will there any problem at the airport? Which one true in First law of thermodynamics:    $Q = \Delta U \pm W = \Delta U \pm p\Delta V$? (where $\Delta U$ is change of internal energy, $W$ work made by system and $Q=cm\Delta T $ heat made by system.)    or $\Delta U= \Delta Q + \Delta W $? (where $\Delta U$ is change of internal energy, $W$ work made by system and $Q=cm\Delta T $ heat made by system.)   First one from one the finnish physics texbook another is from . How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$  I tried to prove that by proving that the following summation in in absolute value converges so the original one converges too, but that's incorrect.  Any other ideas?