Let $S= \{ 1,2,\ldots,n \}$. and $V$ be the set of all functions $f: S \to \mathbb{R}$. $V$ is a vector space defined by: $$(f+ g)(x) = f(x) + g(x) \text{ and } (cf)(x)=cf(x). $$ Find a basis for $V$ and $\dim V$.  I'm not sure how to answer this question, it seems different than how I learned solving a basis, which was through a set of vectors, anyone know how I can tackle this?
Let $V=\{f : S \to \mathbb{R} \}$ be the set of all the functions $f : S \to \mathbb{R}$. Let $S=\{ 1,2,...,n$}, now my train of thought was that the basis of V must be a linearly independent set that Span V so somehow show that there is a set of functions on S that can compose as a linear combination and map to any number in $\mathbb{R}$. Any idea how to get a general answer for basis for dimension of V?
How can I determine file type without an extension on Windows? Missing whitespace in beta site statistics box How to remove desktop environments? I installed few environments that I wanted to try out on Ubuntu 12.04, but none of them worked at all. It could be that I installed them all at the same time, meaning the OS didn't get a chance to work everything out, but either way, they didn't work. I would now like to remove them.   The one's that I installed are Xubuntu/Xfce, Lubuntu/LXDE, Kubuntu/KDE, Fluxbox, OpenBox, cinnamon, and MATE. I installed them. Everything seemed to have been working properly until I actually tried to use the shells and nothing loaded at all. Except for Fluxbox, I think that one worked. I want to know of any way to repair or perhaps just remove the packages entirely. I might have already removed them because I did the apt-get remove command on all of them, but they were still in the list on the login screen. PGFplots foreach equivalent to TikZ's with multiple variables separated by a slash Let $X$ be simply connected and let $ \tau $ be a cycle in $X$. If $ \tau(0)=\tau(1)=x_0$ and $\gamma $ is a path between $x\in X$ and $x_0$, is $\tau$ homologous to $ \gamma\tau\gamma^{-1} $?  I think it is, but I'm not sure about the proof. Can I use the fact that $ \gamma\tau\gamma^{-1} $ is homologous to $ \gamma+\tau+\gamma^{-1} $, where $+$ is the operation between 1-chains (that is abelian)? Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field? In a preadditive category, products are coproducts $\renewcommand{\phi}{\varphi}$ $\require{AMScd}$ My question is different from .  I want to see how in a preadditive category $\mathcal{C}$ a product is a coproduct, following . I'll write the proof up to the point where I get stuck.  Let $A=A_1,B=A_2$ be objects and suppose their product $P$ does exist, with projections $\pi_i:P\rightarrow A_i$. By considering a wedge  $$\begin{CD}A @&lt;{id_A}&lt;&lt;  A @&gt;{0_{AB}}&gt;&gt;  B\end{CD}$$  and the obvious analogous one with $A$ and $B$ exchanged, where $0_{A,B}$ is the zero morphism, we see that the product property implies the existence of two unique morphisms $k_i\in\mathcal{C}[A_i,P]$ s.t.  \begin{align*} \pi_j\circ k_i=\begin{cases} \mathrm{id}_i &amp;\text{if }i=j\\ 0_{A_i, A_j} &amp;\text{if }i\neq j \end{cases}. \end{align*} Using these equations and the definition of zero morphism, it's easy to see that the morphism $m\equiv k_1\circ \pi_1+ k_2\circ \pi_2$ satisfies $\pi_i\circ m=\pi_i$.   This is where I get stuck: In the linked proof, they now state that this implies $m=\mathrm{id}_P.$ But I don't see why this should be true.   Specifically, if $\phi\in\mathcal{C}[S,P]$ for some arbitrary object $S$, it is not apparent why $m\circ\phi=\phi$. All that we know is that there are some morphisms $x_i\in\mathcal{C}[S,A_i]$ that factor through $P$ via the universal property, but these are not necessarily unique.  And we learn even less about $m$ by looking at morphisms in $\mathcal{C}[P,S]$.  What am I missing here? I know I need to have (although I don't know why) a GROUP BY clause on the end of a SQL query that uses any aggregate functions like count, sum, avg, etc:  SELECT count(userID), userName FROM users GROUP BY userName   When else would GROUP BY be useful, and what are the performance ramifications? How to hyperlink only the year part when using natbib and hyperref I am using the natbib and hyperref packages:  \usepackage{natbib}  \usepackage[pagebackref=true, plainpages=false, bookmarks, bookmarksnumbered, colorlinks, linktocpage=true, linkcolor=blue, citecolor=blue, filecolor=black, urlcolor=blue]{hyperref}   With my current settings (above) both the name and the year are hyperlinked.  My question is, whether there is a way to hyperlink (or color) only the year-not the author(s).   I have the impression reading this post  that biblatex does it by default. However changing to biblatex is not an option for me at this point.  Any helpful comments will be greatly appreciated. Limit of derivative is zero. Does it imply a limit for f(x)? Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even Can trusted users be trusted to suggest tag synonyms without a score of 5 in a tag? (on smaller sites, not Stack Overflow) This has been asked before but I feel the original question was more focussed on Stack Overflow than the Stack Exchange network at large.   Related:   Since there are now a lot of sites in the network and many of them use the tag system for categorising questions based on the title of something (for example, Arqade (Gaming), Movies &amp; TV, Sci Fi &amp; Fantasy, etc) I feel that the required knowledge of the subject matter the tag is representing is less of an issue than it perhaps is on other Stack Exchange sites.   In the below screenshot, for example, I shouldn't really have needed a score of 5 in either of the tags to perform what is essentially a cleanup process. I could of course edit all of the posts to delete one tag, so why can't I suggest it as a synonym?    This could be implemented as a per site setting so that it can only be enabled on certain sites if need be.   Can trusted users be trusted to suggest tag synonyms without a score of 5 in a tag? What determines the direction of precession of a gyroscope? I understand how torque mathematically causes a change to the direction of angular momentum, thus precessing the .  However, the direction, either clockwise or counterclockwise, of this precession seems to me a bit arbitrary beyond mathematical definitions. (What if torque were defined by a "left hand rule", for example?).  What's the fundamental physical reason that the precession of a gyroscope would be directed in a certain direction? I am using wp_get_attachment_image_src to construct the srcset attribute of an img tag. In that attribute I want to output a comma separated list of image urls along with the image widths.  Using the Media Settings page, I have configured the image sizes to be as follows:   Thumbnail size: 150x150 Medium size: 640x480 Large size: 1024x768 Small size: 278x320   The wp_get_attachment_image_src function returns the correct urls and the correct widths for the image sizes full, medium and small. small is a custom size I added using add_image_size.  If I call wp_get_attachment_image_src( $image_id, 'large' ), the returned array looks like this:  array (     0 =&gt; 'http://localhost/wp-content/uploads/2014/10/image-1024x768.jpg',     1 =&gt; 640,     2 =&gt; 480,     3 =&gt; true, )   Checking the image which can be found at that url turns up the image with the correct dimensions.  Currently i am using the php builtin function  to circumvent this problem, but this function call takes significantly longer than simply accessing the index returned by wp_get_attachment_image_src:  Without getimagesize (mtime before and after execution):   '0.83761000 1415195440' '0.83831600 1415195440'   With getimagesize:   '0.34808000 1415195635'  '0.35323900 1415195635'   What can I do to make wp_get_attachment_image_src return the correct dimensions? What is the proper way to write the plural of a single letter? Or a word used as a word ('mention' usage)? (another apostrophe question) When writing (a blog post, script, etc..) what is the proper way to indicate two or more instances of a single letter? For instance, in Monty Python's Bookshop Sketch:     C: I wonder if you might have a copy of "Rarnaby Budge"?      P: No, as I say, we're right out of Edmund Wells!      C: No, not Edmund Wells - Charles Dikkens.      P: (pause - eagerly) Charles Dickens??      C: Yes.      P: (excitedly) You mean "Barnaby Rudge"!      C: No, "Rarnaby Budge" by Charles Dikkens.  That's Dikkens with two Ks, the      well-known Dutch author.      P: (slight pause) No, well we don't have "Rarnaby Budge" by Charles Dikkens      with two Ks, the well-known Dutch author, and perhaps to save time I      should add that we don't have "Karnaby Fudge" by Darles Chickens, or      "Farmer of Sludge" by Marles Pickens, or even "Stickwick Stapers" by Farles      Wickens with four M's and a silent Q!!!!!  Why don't you try W. H. Smith's?      C: Ah did, They sent me here.   I had always believed that plural never uses an apostrophe before the 's' (it's only used for possession), but I have rarely seen in written material the format "four Ms". (On a side note, whoever wrote  also used "two Ks".)  On a side note, and perhaps this should be a separate question, if a Compact Disc is a CD, then two Compact Discs would be two CDs right? (I see "CD's" written everywhere) Experimental data for asymmetric Newton cradle Using a "successive impact model" (as if each ball were seperated from the other ones), I produced the following animations:    You can see any combination of balls with masses of 1 or 2 (left) or 1 and 4 (right).  Unfortunately, I do not have any Newton cradle to make some experiments, and I'd like to compare my results with observations. I contacted the author of , which is the most instructive one I found about asymmetric Newton cradle. In particular, he writes that the "OoO" configuration behaves as the "ooo" one, which is not what I simulate. But the author is not sure whether who is wrong because he did the experiments a long time ago.  I am especially interested in asymmetric Newton cradle with simultaneous impacts: left and right balls collinding at the same time.   Please let me know if you have any idea on where to find such experimental data.  Edit Just of few general remarks to avoid extending the comments.    Energy and momentum are conserved. However, they are sufficient to ensure uniqueness of the post-impact velocities only when there are 2 balls. The successive impact model precisely consists in propagating the impact ball after ball, and therefore leads to a unique solution (which conserves momentum and energy globally). From the waves point of view, changing the mass of a ball can be done by increasing the diameter, or the density (or a combination of both). This simple model cannot account for such subtleties of course, but maybe it does not matter when the balls are "small enough". Even if I think it's quite basic, following WetSavannaAnimalakaRodVance's comment, I am ready to hand out my Mathematica code to anybody who wants it. I'll check for good ways to share it. What's the best way to extend the User model (bundled with Django's authentication app) with custom fields? I would also possibly like to use the email as the username (for authentication purposes).  I've already seen a   to do it, but can't decide on which one is the best. There are many collectable flags throughout the game of Assassin's Creed and every time one is collected, you get a progress indicator (e.g., 33 of 100).  Is there any reason to collect these, other than bragging rights?  I recall getting all available flags in one particular town and never saw any special reward or secret revealed. I would like to build a new computer (it's been 7 years since the last one). Does Windows 10 support more than one ? Export EDITOR with a flag I'm using Zsh and some CLI's such as Google's gcloud load files via $EDITOR.    For me, I have set this as:  export EDITOR="/usr/local/bin/mvim"   With aliases for that editor using the -v flag which loads vim in the terminal:   mac_vim="$EDITOR -v"  alias vi=$mac_vim  alias vim=$mac_vim  alias v=$mac_vim   That works but when the gcloud CLI calls $EDITOR it does not have the -v flag so it opens in a new terminal.  I thought I could change that by instead adding the -v flag here:  export EDITOR="/usr/local/bin/mvim -v"   However, when I try and run that on the command line I get:  $EDITOR text.html zsh: no such file or directory: /usr/local/bin/mvim -v   Is it possible to include a flag when exporting my $EDITOR?