I wish to prove that the dimension for the set of all linear mappings $L:V \to W$ is equal to $\dim V \times \dim W$. I know that any general linear mapping can be represented as a matrix, so intuitevely it makes sense that the dimension should be $\dim V \times \dim W$ but I cannot find a way to prove this mathematically
I have found that the question I want to ask someone had asked, here is the website:    Here is my question: Why $\dim(\hom(V,W))=\dim V * \dim W $?    Thanks for explanation~!
Let's get rid of the 10K flag queue are pretty cool... You get a birds-eye view of activity on the site, a &quot;dashboard&quot; view of what's happening. Some of the individual tools haven't scaled particularly well with Stack Overflow's growth, but the concept behind them is still sound: we trust you to enough to be a bee watcher now.  ...and then there's . What a let-down! Once upon a time, this queue contained spam and offensive flags, which one might reasonably assume were important enough to put in front of the more trusted members of the site. Re-flagging them brought them that much closer to deletion, while editing them offered salvation to some hapless author. Nowadays, it's a bunch of Not an Answer flags and a smattering of assorted cruft. 10K users can't even vote to delete these; only 20K users have that privilege. Re-flagging them does nothing but increase their priority in the moderator flag queue, where they frequently outrank more pressing issues;  has . Also, it's , and the behavior has diverged far enough from that of the moderator flag queue on which it is based that it has become an active hindrance to further development of the tooling there. Worst of all, we have  that's available to anyone with the editing privilege. It even has logic built in to prioritize answers likely to be deleted for users with . It's like we gave you a car for your 10th birthday, and then replaced it with a rusty bicycle when you turned 16. It was a nice idea, but it has outlived its usefulness. Proposed changes The /tools/flagged route goes away. Period. No replacement. The rest of the 10K tools stick around as informational pages. Not an Answer flags go into /review/low-quality, . Then we beef up the Low Quality review process to make better use of more experienced reviewers and solve this whole &quot;declined / helpful / disputed&quot; flag debate once and for all:  Effective # of reviews required == ReviewsRequired + # of applicable flags (where ReviewsRequired is 2 on Stack Overflow, 1 everywhere else). So 1 VLQ or NAA flag means EffectiveReviewsRequired=3. LQ tasks are not dequeued until one of the following conditions is met:  Post is edited from within review. Outcome: flags are marked &quot;helpful&quot; (current behavior). Post accumulates 3 Delete votes (can only happen when post scores &lt;= 0 and reviewers have &gt;= 20K rep). Outcome: post is deleted, flags are marked &quot;helpful&quot;. Task accumulates EffectiveReviewsRequired &quot;Looks Good&quot; reviews. Outcome: if the number of (Recommend)Delete reviews is &gt;= the number of Looks Good reviews, mark flags &quot;disputed&quot; and raise DisputedLowQuality mod flag. Otherwise, mark flags &quot;declined&quot;. Task accumulates 6* RecommendDelete + Delete reviews. Outcome: mark flags &quot;helpful&quot;. If the post scores &gt; 0 then raise DisputedLowQuality mod flag, else just delete post (current behavior).   As under the current system, flags on posts that've already completed one full review cycle without being deleted should skip /review and go directly into the mod queue.  End result  Flags get handled faster, more accurately, and with less wasted effort from 10K reviewers. Moderators are free to focus on situations that can't be handled by the community - the ! Developers are free to make enhancements to the moderator tooling without having to work around 10K user restrictions.  Questions Am I forgetting anything here? I haven't really spent much time in the 10K flag queue since back when it was filled with spam flags and therefore useful - is there a use-case that you depend on that would be lost with its removal? Post 'em below. *: on Stack Overflow, only . I'm currently working on the following question:  "At least one of Plato and Democritus believed in the theory of forms. Plato believed in the theory of forms only if he was not an atomist, and Democritus was an atomist only if he did not believe in the theory of forms. Democritus was an atomist. Prove, using Natural Deduction, that Plato was not an atomist"  I have been working for quite some time to take the information and turn it into a set of literals and statements I can work with.  So far, I have got $P \lor D$,  $P \Rightarrow \lnot a$,  $D \Rightarrow A$,  $D = A$  However, I cannot seem to make these prove the statement using natural deduction, any hints? Say a security has been trading at under $20.10, and there are limit orders selling at $20.10 and $20.21. Then Simon issues a limit order to buy at $20.21. According to , Simon would end up buying shares at $20.10.  Why does the exchange favor the buyer here?  I.e., why does the transaction execute at the seller's limit of $20.10 instead of the buyer's limit of $20.21?  (I.e., why don't Inara/River get to sell at $20.21 / above their limit, but Simon gets to buy at $20.10 / below his limit?)  Is the example flawed?  If it makes a difference, I'm mainly curious about NASDAQ and NYSE.  I tried Googling for specs (e.g. ), but couldn't find anything.  Only found that some  implementation executes the average of the buy/ask.  Just curious how the system works. Flair: 500 Internal Server Error - http://stackexchange.com/error?aspxerrorpath=/users/flair/378431.png I installed Ubuntu yesterday, so I'm trying to get used to it.  I'm really close to my router, but my wifi signal gets too low. Sometimes I'm in Facebook or Youtube and the signal is high, almost full, but it is usually in one or two bars, and sometimes none.  I can use Internet normally, I can watch videos and stuff, but the problem is when I try to update something or Ubuntu is downloading stuff, the downloads get stuck or get cancelled telling me that I should check my connection.  How can I fix this? I should be getting a really good signal, I'm really close to my router, my phone has full signal, and a notebook with Windows 7 has full signal here too.  Sorry my bad English and I hope you can help me.  $ lsusb Bus 009 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 004 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 008 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 007 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 003 Device 002: ID 0bda:8179 Realtek Semiconductor Corp.  Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 006 Device 002: ID 0458:003a KYE Systems Corp. (Mouse Systems) NetScroll+ Mini Traveler / Genius NetScroll 120 Bus 006 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 005 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub How should I install HPLIP binary/proprietary plugin/driver for my HP printer or printer+scanner? I apologize if this question is posted on the wrong portion of the website, but I knew no better place to ask.  I've come to see that some effects, such as super-conduction may happen at either extremely low temperatures, or at extreme temperatures (the recent discovery of superconducting iron in the bowels of the earth, for one), and this is fascinating.  My question is: can (does?) the same principle be applied to quantum effects? Would it be applicable to, for example, say that although quantum effects happen at nano scales and below they may happen on a grand scale of say, the entire Universe (as a solitary unit)?  Can said quantum effects happen at moments of say, picoseconds after the Big Bang due to the turbulence of forces involves in such an event? Or even within the core of a dying star?  Thank you for any answers or clarification of my misinformation. I noticed relation between modulo operation and number which is power of two  Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to  $ ( 3431242341 \mod 2^9 ) \mod 2^5 $   I tried many examples and it seems to be true in general, and I am not sure if it is a coincidence or true in general that I can use first modulo operation ( greater number) and the result will be the same. Difference between regex [A-z] and [a-zA-Z] How to read command line arguments of another process in C#? How do I create a villager with custom trade offers? You can reach this IP address  (if my computer is online). I want to assign a domain to it so I wonder how is that possible when I have no DNS? I do own my domain which is ".com". My IP is static.  Thanks Thailand visa exemption: any rule limiting number of entries per year? How should you interview for QA *Automation* positions? How to execute sudo commands with Expect & send commands in bash script? What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$? I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.:  $$\cos(z)= \cosh(iz)$$ $$\sin(z)=-i\sinh(iz)$$  where $z$ is a complex number.  From a geometric point of view, what I understand is that cos is the composition of a rotation through $\frac{\pi}{2}$, followed by cosh, and sin is the composition of a rotation through $\frac{\pi}{2}$, followed by sinh, followed by a rotation through $-\frac{\pi}{2}$ (where sin, cos, sinh, cosh are defined as complex functions).  Where does this connection come from? Is there some way it can be visualized in terms of complex mappings? (I'm not asking for a proof of the identities, I already know one). Mid-1980s animated theatrical release Though at the time I saw it I did not know it, this would be called "anime" today. It was in English, and I don't remember if the characters lips matched the audio track. I couldn't have been older than about 3rd grade, so that puts it around 1985ish I believe.  It was released theatrically in the United States, I do not know if it was a limited release. I saw it in Cincinnati, Ohio. No idea what movie theater chain.  The story was rather typical, young man who goes on the hero's journey. Definitely science fiction elements (flying vehicles, maybe spaceships). He comes across or is given a sword hilt without a blade. He learns to use it, and when he does it acts as if the blade is there. At the finale, he has lost the hilt entirely and, thinking that he has lost, almost gives up. But a (voice in his head, his mentor, can't remember what) something tells him to try despite this, and it's as if he has the sword even without the hilt. He then triumphs.  Can anyone identify this movie? One of the products being shown a CES is a  intended to let you charge a tablet from a laptop USB port that normally wouldn't work because it didn't provide enough power.  The CNET writeup implies that it's actually boosting the power output from the port beyond the normally available amount.  This seems improbable to me because the ports should have current protection to protect themselves from shorts.  Also, if it was possible to bypass the current limit externally, I'd've expected at least some tablet vendors to have included the ability to do so directly in their cables.  One possibility that occurs to me is that the dongle is just spoofing responses to the tablet when it queries the laptop about the maximum power it can deliver while only providing a trickle charge that the tablet would otherwise reject as too slow to bother with. How to cut the non-overlapping part of 2 shapes in Illustrator CS6 Which of the following is most correct?      The distance of the shortest path must not be higher than 10km.   The distance of the shortest path must not be greater than 10km.   The distance of the shortest path must not be bigger than 10km.   Or is there an even better word?