Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.
Why $\dim U+\dim U^\perp=\dim V$?
Find four groups of order 20 not isomorphic to each other and prove why they aren't isomorphic.   So far I thought of $\mathbb Z_{20}$, $\mathbb Z_2 \oplus\mathbb Z_{10}$, and $D_{10}$ (dihedral group), but I can't find another one. Would $U(50)$ work? I know it has order 20 and is cyclic but I'm not exactly sure how to move from there. Can someone to point me on the right direction? the verb in plural with one company ST_Length (GEOM, true) for calculating distances in Geographic Reference System I have seen several answers and proofs here at MSE that a self-adjoint operator $T$ on a Hilbert space $H$ where $T^2=T$, implies $T$ being an orthogonal projection operator.   The thought struck me; what would happen if the condition $T^2=T$ is replaced by $T^3=T^2$ or instead by $T^4=T^3$ or why not more generally by $T^n=T^{n+1}$ for some $n\in\mathbb{N}$? Would this still imply $T$ to be an orthogonal projection operator?  I guess someone must have thought of this before me? In that case perhaps there is a reference in the literature for this or perhaps the proof is way simpler than I might expect? Poincaré's theorem about groups Footnotes in tables? Is there a word to describe the virtue of "to do something and ask for nothing in return"? I am wondering if there is a positive, formal counterpart to the slang "No strings attached" to describe, for example, a parent's unquestioned devotion to their children, or the virtue of someone's "a friend in need is a friend in deed", help first, question later mentality. Show that $2730\mid n^{13}-n,\;\;\forall n\in\mathbb{N}$   I tried, $2730=13\cdot5\cdot7\cdot3\cdot2$  We have $13\mid n^{13}-n$, by Fermat's Little Theorem.  We have $2\mid n^{13}-n$, by if $n$ even then $n^{13}-n$ too is even; if $n$ is odd $n^{13}-n$ is odd.  And the numbers $5$ and $7$, how to proceed? How do I remove old kernel versions to clean up the boot menu? Preloading images with jQuery I'm looking for a quick and easy way to preload images with JavaScript. I'm using jQuery if that's important.  I saw this here ():  function complexLoad(config, fileNames) {   for (var x = 0; x &lt; fileNames.length; x++) {     $("&lt;img&gt;").attr({       id: fileNames[x],       src: config.imgDir + fileNames[x] + config.imgFormat,       title: "The " + fileNames[x] + " nebula"     }).appendTo("#" + config.imgContainer).css({ display: "none" });   } };   But, it looks a bit over-the-top for what I want!  I know there are jQuery plugins out there that do this but they all seem a bit big (in size); I just need a quick, easy and short way of preloading images! How to pronounce "Stannis's"? I know that the &quot;s&quot; becomes &quot;z&quot; when it comes after a voiced sound. And it is pronounced as &quot;s&quot; when it comes after a voiceless sound. I also know rules about possessives but I don't know how natives pronounce a word when it ends in S and &quot;apostrophe S&quot; is added to it. For example: &quot;Stannis's army&quot;. Do you add another syllable to it? Like Stannisiz? or just Stannis and ignore the possessive S? I really have difficulty in pronouncing words that end in S and possessive S is attached to it. How does Motherboard and CPU affect RAM choice? I have installed bootable ubuntu on hard disks but now I can not format them with Disk Utility program. When I try to format it says "One or more partitions are busy on /dev/sdb" for both hard disks. Is there any way to format these drives?  EDIT: With "Disk Utility" program when I select one of the hard drives It shows 3 volumes;  99 MB FAT 111 GB EXT4 9 GB UNKNOWN   I can not format this unknown volume.  With Gparted program, It doesn't shows Unknown volume only shows 111GB EXT4. Set Snap Mode with PyQGIS So, I'm running linux Mint (but I guess the same applies to Ubuntu) and as is common knowledge, their TeX distribution tends to be slightly outdated. Now I can't have none of that and decide to install it with the install-tl from the TeXlive homepage. Somehow, I cannot install it as a normal user (this I find normal as it is installing in the root directory), so I proceed to install TeXlive   sudo perl ./install-tl   and add the paths to .bashrc. All is well and my TeXlive installation works. Except for the texhash option. Running it as a normal user is no problem, except that I installed TeXlive as root, and so running texhash as normal user doesn't do anything (directory not writable. Skipping...). No problem I think, I'll just do it as root. But somehow texhash isn't installed for root.  Now, texhash is somehow connected to updating the directory tree, but I don't really know what it does. I use it on my ~/texmf/ folder to add my packages to a global path (I think), but when looking at info for this question I see in a comment for  question that this is apparently not needed, though I don't understand what I'm to do now.  So if I'm not adding new packages from CTAN or updating, I really don't need texhash and all is good?   To summarize: I'd like to know:    Why should I need texhash? Can I avoid installing as root on linux Mint/ubuntu ( they say you don't have to be root, that it's even discouraged)? Should I somehow install texhash for root?   I realize this might be a vague question at best, but I didn't find an answer I readily understood, so I'd appreciate any thought on this. First post review queue: answers are questions, questions are answers I believe there is a typo in the  review queue. When reviewing a question, the information message asks the user to review the answer. Take a look at the image below.    Similarly, when an answer is posted in the queue, the description box calls it a question instead. Has anyone else noticed this? Where does gravity get its energy from? Why does autoboxing make some calls ambiguous in Java? I noticed today that auto-boxing can sometimes cause ambiguity in method overload resolution. The simplest example appears to be this:  public class Test {     static void f(Object a, boolean b) {}     static void f(Object a, Object b) {}      static void m(int a, boolean b) { f(a,b); } }   When compiled, it causes the following error:  Test.java:5: reference to f is ambiguous, both method     f(java.lang.Object,boolean) in Test and method     f(java.lang.Object,java.lang.Object) in Test match  static void m(int a, boolean b) { f(a, b); }                                   ^   The fix to this error is trivial: just use explicit auto-boxing:  static void m(int a, boolean b) { f((Object)a, b); }   Which correctly calls the first overload as expected.  So why did the overload resolution fail? Why didn't the compiler auto-box the first argument, and accept the second argument normally? Why did I have to request auto-boxing explicitly? Finding a tricky limit Let $t∈\Bbb N$ and let $x=6t+1, \:y=12t+1$ and $z=18t+1$. $x, y$ and $z$ are all primes and let $n=xyz$. Prove that  $a ^{n-1} \equiv 1 \pmod  n\;$ whenever $ a∈\Bbb Z$ and $\gcd(a,n) = 1$.  I have started using the Euler-Fermat Theorem since $\gcd(a,n)=1$. Giving $\phi (n)=(1296t^3) $ But unsure how to proceed?