As the title says: Prove that any two real finite dimensional vector spaces with the same dimension are isomorphic to one another?  How would you approach this proof?
Given the following problem     Let $U$ and $V$ be finite dimensional vector spaces, such that $\dim(U) = \dim(V)$. Prove that $U$ and $V$ are isomorphic.   I was just wondering if somebody could critique and validate my proof.  Proof  Two vector spaces $U$ and $V$ are isomorphic if and only if there exists a bijection $\phi: U \to V$. That is, an invertible linear map between the two. Let $\dim(U) = \dim(V) = n$ and let $\left\{u_1, \dots, u_n\right\}$ and $\left\{v_1, \dots, v_n\right\}$ be bases for $U$ and $V$, respectively. It follows that any vector $u \in U$ can be written as  $$ u = c_1u_1 + \cdots + c_nu_n $$  We define $\phi: U \to V$ as  $$ \phi(c_1u_1 + \cdots + c_nu_n) = c_1v_1 + \cdots + c_nv_n $$  Observe that $\phi$ is invertible. That is, given a vector  $$ v = d_1v_1 + \cdots + d_nv_n $$  in $V$, we define $\phi^{-1}: V \to U$ as  $$ \phi^{-1}(d_1v_1 + \cdots + d_nv_n) = d_1u_1 + \cdots + d_nu_n $$  Since there exists a bijection between $U$ and $V$, they are isomorphic.
There are several spells that require the target to be "humanoid" such as , , etc.  However, if the target is Polymorphed after that spell has been successful, then what happens?  Does the change in form to Beast end the spell effects? Does it suppress the spell until the correct form returns? Does it do something else? I am compiling the following MWE using XeLatex within TeXnicCenter and get a garbled equation number:  \documentclass{article} \usepackage{amsmath} \usepackage{polyglossia} \setmainlanguage{hebrew} \newfontfamily{\hebrewfont}{Arial}  \begin{document} \begin{equation}\label{eq:1} 2+2=4. \end{equation} \end{document}      What is going on here? What does the symbol $\subset\subset$ mean? In some texts (mainly complex analysis or harmonic analysis) I sometimes see the following double subset symbol $\subset\subset$ for inclusion relation of two regions, e.g., $\Omega$ and $\Omega'$ are two regions in $\mathbb{C}$ such that $\Omega \subset\subset \Omega'$. I never figured out what it means exactly; I always interpreted it as the closure $\overline{\Omega}$ is contained in $\Omega'$ (so that some nasty boundary effects can be avoided). Is that right? Or does $\subset\subset$ mean some other kind of inclusion? Thanks. Windows 10 64-bit requirements: Does my CPU support CMPXCHG16b, PrefetchW and LAHF/SAHF? I'm currently trying to find out whether or not it would be a good idea to update my slightly dated notebook (Windows 7, 64-bit) to Windows 10. The problem is that Microsoft states in their  that to use the 64-bit version of Windows 10, a processor which supports CMPXCHG16b, PrefetchW and LAHF/SAHF is needed. I know that my processor is 64-bit (Intel Core i5-2430M @ 2.40 GHz) but I don't know how to find out whether it supports CMPXCHG16b, PrefetchW and LAHF/SAHF, and I have no idea what these mean.  I tried using Google and got no results that made sense to me, only some comparison of my i5 and a similar AMD processor. I also checked Intel's ARK website. If somebody could point me to where I can get more info I would be really grateful. Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$ I want to set up a diary that compiles in two different orders: One is work by day, the other is work by project according to tags I give to the particular chunks of work I did each day.   For example, I write my diary as usual in work by day form:  2/2/15  -- bought a new screwdriver (tag: car maintainence)   -- scetched my plan for world domination (tag: world domination)  3/4/15  -- started studying physics (tag: world domination)  -- finished tax declaraction (tag: burocracy)   Compiling by project would yield tags as sections:  car maintainence    -- 2/2/15: bought a new screwdriver  world domination  -- 2/2/15: scetched my plan for world domination  -- 3/4/15: started studying physics  burocracy  -- 3/4/15: finished tax declaraction   I know this is pretty close to the literal programming paradigm the whole LaTex documentation follows: A code chunks in a document can be compiled to two different orders: For human understanding and for use by the machine. However, I want two human readable versions --say pdf-files-- as output. Is there some package that helps? Has there been any progress made on granting a developer account for SF and HubExchange Devs? It feels conceptually wrong to me a SF product doesn't allow developers to perfect and develop a wider solution offering without paying +10K license fee per year. Even without CRM the HubExchange market could be a richer portal for consumers and developers if we were allowed a means to test our concept.  I think we'd all be perfectly happy with an account that was blackholed or limited in such a way where we could only send a finite qty of emails to an email address associated to the user.  If there's a forum or avenue where we can assist in raising the value proposition please let us know as I know there are many people looking. Many people want to try/use Android without buying a new Android phone.  How can I install Android on my current phone or device running Nokia Symbian, Maemo, or MeeGo; Microsoft Windows Mobile or Windows Phone 7/8/10; Apple iOS (iPhone); RIM's BlackBerry OS; Palm's WebOS; Samsung's Bada; HP's WebOS; Mozilla's Firefox OS; or other phone / operating system? How can I stop Ubuntu Dock from extending to the edges? By default Ubuntu Dock in Ubuntu 17.10 and later releases always extends to the edges (top-bar-to-bottom if placed on the left or right and left-to-right when placed at the bottom) even when it's not filled with icons. Is there a way to prevent this behaviour (i.e. I want to make the dock-length flexible so that as the number of application icons in the dock increases, it should expand accordingly)?  ➡️➡️➡️  In other words, is there a way make Ubuntu Dock look like a &quot;proper dock&quot; instead of a side/bottom panel with full-height/width? BeanFactory vs ApplicationContext I'm pretty new to the Spring Framework, I've been playing around with it and putting a few samples apps together for the purposes of evaluating Spring MVC for use in an upcoming company project. So far I really like what I see in Spring MVC, seems very easy to use and encourages you to write classes that are very unit test-friendly.  Just as an exercise, I'm writing a main method for one of my sample/test projects. One thing I'm unclear about is the exact differences between BeanFactory and ApplicationContext - which is appropriate to use in which conditions?  I understand that ApplicationContext extends BeanFactory, but if I'm just writing a simple main method, do I need the extra functionality that ApplicationContext provides? And just exactly what kind of extra functionality does ApplicationContext provide?  In addition to answering "which should I use in a main() method", are there any standards or guidelines as far as which implementation I should use in such a scenario? Should my main() method be written to depend on the bean/application configuration to be in XML format - is that a safe assumption, or am I locking the user into something specific?  And does this answer change in a web environment - if any of my classes needed to be aware of Spring, are they more likely to need ApplicationContext?  Thanks for any help. I know a lot of these questions are probably answered in the reference manual, but I'm having a hard time finding a clear breakdown of these two interfaces and the pros/cons of each without reading thru the manual with a fine-tooth comb. Digits of $\pi$ forming primes? Let $f(n)$ be the first $n$ digits of $\pi$ . How many times is $f(n)$ prime on the interval $[1, k]$ ? Are there infinitely many prime $f(n)$'s on $[1, \infty)$?  I know this is probably be a very hard question and most likely an open problem, but I am just curious. If you are reading this, you may have been affected by one of our post quality bans. If so, it is important that you  to understand what has happened and how to regain the permissions that you lost. While trying to ask a question, one could see:  We are no longer accepting questions from this account. See  to learn more.  Likewise, for answers:  We are no longer accepting answers from this account. See  to learn more.   Why am I getting this message? Are deleted posts taken into account too? Is a question/answer ban the same as a suspension? How do I avoid getting a question ban? How long do I have to wait before I can post again? What can I do to release the ban? How can I reactivate my account? Can I simply create a new account? I'll just ask somewhere else on the SE network, and they'll migrate my question to the correct site! Will a ban on one Stack Exchange site affect my standing on other sites in the network? Does this apply to meta sites too? My account is in good standing. Why am I still blocked? I have found film reels some with tails of film others not. Is there a way of telling if they have been exposed/used in a camera? Film types are for 35mm and one Kodak Advantix. Acknowledgements outside of document environment How do I add an acknowledgements section (maybe not specifically \section) outside of (before) my document environment?  in my .cls, there is  \def\@acknowledge{} ... \def\acknowledge#1{\booltrue{@ack} \gdef\@acknowledge{#1}} ... \def\ackpage{ \chapter*{Acknowledgments} \@acknowledge \newpage }   but it seems that I find problems when I put newlines in the  \acknowledge{folks in here}  (see  ) due to versus as a result of Why must a planet be in a star's "habitable zone" to have life? I just would like to know if a question or an answer saved draft is visible to someone else, apart from the OP.  Will a copy of a saved draft remain visible even after it has been deleted? Who has access to it? pkgloader is not finding l3regex How can I convert JPG into PDF easily? Table cells larger than they are meant to be I've created a map system for a game that runs on the principle of drawing the picture of the map from tiles. There are many reasons for this which I won't go into here but if you really want to know then I'm sure you can find out how to contact me ;)  I have  so you can see exactly where the problem lies and the source. The issue is the line between the top 2 tiles and the bottom 2 tiles, I can't figure out why it's gone like this and any help would be appreciated.  In the source is a marker called "stackoverflow", if you search for "stackoverflow" when viewing source then it should take you to the table in question.  I have also uploaded an .