Let $V_1,V_2$ be vector space and $\{e_i\},\{\overline e_i\}$ are basis respectively. $\forall ~l\in L(V_1,V_2; F)$ ,why $l$ can be represented as  $$ l=\sum\limits_{ij} a_{ij} \omega^i\otimes \overline\omega^j  ~~~~~~~? $$ $a_{ij}=l(e_i,\overline e_j)$ and $\{\omega^i\},\{\overline\omega^i\}$ is dual basis of $\{e_i\},\{\overline e_i\}$ . $L(V_1,V_2; F)$ is 2 multilinear function space .
Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck.
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Of course this doesn't 100% guarantee that someone won't get it wrong on occasion. But this is an awful lot of in-your-face guidance without simply telling people "read our FAQ" (which they never do). And if a user manages to step over all that guidance and still manages to misuse meta, I have no compunction about summarily removing a question they should never have posted.   But this just-in-time learning is a very effective way of learning a new site in a gentle and thoughtful way. If this is abused more often than not, it is easy to disable. But it's better than assuming clueless newbie every time. B2B application: users has its own users B2B Scenario I am trying is that:  The following Roles exists in the system:  Super Admin Super Admin Staff Owner (client) Admin (owner i.e. client and creates admin to define sales counters) Sales Counters  That means there are users who'll create their own sub users e.g.  [FLAG-A] ROLE &quot;SUPER ADMIN&quot; creates a ROLE &quot;Owner&quot; and &quot;OWNER&quot; creates its own sub users ROLES i.e. &quot;ADMN&quot; &amp; &quot;SALES COUNTER&quot;. Each role obviously has its info to login to the system I tried to design database as following:  In this diagram I am assuming that main user creates sub user that in my view many to many relationship so I have to add table between them as &quot;USER_PERSON&quot;. Problem is to set &quot;WHO CREATED USER&quot; in the system. I mean which user has created sub user as described in [FLAG-A]  Please check this diagram How can I view Google Photos that are not in a Google Photos album? Go to  Add a few photos to a photo album Go back to All Photos   See all of the photos there, even ones you copied to the Album. No problem here, per se. At least not yet.  What if you want to delete photos that you did not copy to that album?  Where is the selector or filter that filters those "in an album already" photos from this main view?  Or is there a way to create such a filtered view such that I can then cherry-pick photos for deletion? Is the force which throws a body on a rotating disk outwards centrifugal force? If some object is kept in a radial groove made on a disk and the disk is rotated with say constant angular velocity.   Now as we know that the object moves outwards ie away from the centre of the disk. I want to know which force is it which throws the object outwards.   As centrifugal force is a pseudo force and hence when the observer is in ground frame it shouldn't be present so which force exactly pulls it out. ls command operating differently depending on recipient permanent hat-like thing? I really like how, in the winter bash, there are hats. They are pretty cool and actually encourage me to get badges, something I didn't care about before. I think there should be something like this, but all year round, to encourage people to get badges. Badges, to me, seem currently pretty useless and I never want to get any because...there is no good reason to. Hats: well, hats are just awesome How to install KDE along Unity? I have just installed the latest Ubuntu 14.04 and I would like to install KDE along side my Gnome/Unity that I have now. How can I do this?  Thanks! Problem: Evaluate $$ L=\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left(  \left\lfloor \frac{2n}{k} \right\rfloor -2\left\lfloor \frac{n}{k}\right \rfloor \right).$$  Please help me with this one.  I tried using the sandwich theorem by first constructing the following  $$ 2 \geq \left\lfloor \frac{2n}{k} \right\rfloor -2\left\lfloor \frac{n}{k} \right\rfloor \geq -1 $$ and then summing over. But the limit of the two bounds are different so sandwich theorem doesn't help!. Is it possible to construct sharper bounds?  Also I observed that if we ignore question of continuity, etc. then we can write,  $L= \displaystyle \int ^{1}_{0}\left\lfloor \frac{2}{x} \right\rfloor -2\left\lfloor \frac{1}{x} \right\rfloor dx$  Now I am able to compute the above integral. But is the above process correct? If then how can make it rigorous? Or if any other approach is possible, please tell. Beta regression of proportion data including 1 and 0 Google AdSense Policy Non Compliant "Site Behaviour: Navigation" How can I make a .jar file executable? I'm trying to run a jar application under Ubuntu, so I installed OpenJDK Java 7 Runtime, but when I open this application I got this message :     The file '/home/aimad/Programms/jMerise/JMerise.jar' is not marked as   executable.  If this was downloaded or copied from an untrusted   source, it may be dangerous to run.  For more details, read about the   executable bit. Can I travel to the USA while working remotely for my non-US employer? I am trying to put a server together to install Ubuntu. I bought an Intel S1200v3RPS motherboard and plugged an Intel Xeon E3 processor into it. I purchased a 1U chassis from Supermicro with a power supply.  The power supply does not have a 5-pin cable to the motherboard's Auxiliary Power Supply socket. A bit of research did not give any definite answer on the need for the 5-pin connection. The motherboard does not power on and the CPU fan is off. I tested the power supply on a desktop Intel motherboard and it works fine.   I would like to buy a new power supply but could not find one which expressly stated that it comes with the 5-pin aux power supply cable. Any recommendations on a power supply unit for this motherboard would be helpful. Any comments on the problem would be greatly appreciated. Turning vector grid into individual shapefiles using QGIS? I am using QGIS and I created a vector grid on a crop field in order to cut it into plots. I want to save each grid cell into individual polygon shapefiles automatically.  Here is an example of a vector grid I would like to split into 117 shapefiles:    Eventually, I did it manually with the selection tool and "Save as" but it is really time-consuming when there are more than 100 plots to save.    Is there any way to do it faster and more automatically?  Each plot as an attribute named "Plot" that can discriminate against them (basically being id + 1). It's the attribute used here to label them. How to get this blur texture effect in photoshop? Having the derivative be an operator What is PECS (Producer Extends Consumer Super)? find and echo file names only with pattern found I used this a lot, the improvement I try to achieve is to avoid echo file names that did not match in grep. Better way to do this?      for file in `find . -name "*.py"`; do echo $file; grep something $file; done What's the 'correct' way of registering/installing an Assembly to the GAC? Can you supply arguments to the map(&:method) syntax in Ruby?