Let $V=\mathbb{M}_{n}(\mathbb{F})$ the space of $n\times n$ matrices. Let $f\in V^{*}$ satisfying $f(AB)=f(BA).,$ So, exists unique $\lambda\in\mathbb{F}$ such that $f(A)=\lambda\textrm{tr}(A)\,\forall \,A\in V$, where $\textrm{tr(A)}$ denotes the trace of $A\in V$.  Any tips to follow will help me. I think that $\lambda$ has something to do with the eigenvalues of $A$, but I don't know.
We know that the trace of a matrix is a linear map for all square matrices and that $\operatorname{trace}(AB)=\operatorname{trace}(BA)$ when the multiplication makes sense.   On the , under properties, it says that these properties characterize the trace completely in the following sense: If $f$ is a linear function on the space of square matrices satisfying $f(xy)=f(yx)$, then $f$ and $\operatorname{tr}$ are proportional. A  of the page gives the justification, but I do not understand the logic of it. Thanks
Convert char to int in C# Why does Google not consider dot in usernames of Gmail addresses? E.g.: If your Gmail ID is abc.xyz@gmail.com, it considers this the same as abcxyz@gmail.com.  Why is this so? How to disable Spotify indicator in the top panel? The Spotify indicator icon clashes with the native Unity theme, and is redundant, since it is already listed in the Sound indicator menu. Is there a way to remove or disable the Spotify indicator? Or at least to change the icon (couldn't grep it on my filesystem). I am using the Spotify "preview" with 12.04. Thanks! How do I put /home in a different partition? Yeah, so I have 2 hard drives, on with 2 partitions(backups and windows), and I have another that also has 2 partitions with only one used. I have Ubuntu on hard drive 2:partition one, but I have come to a point when the /home folder can't hold any more files, but the partition is only about 15% filled up. So I did some researching and figured out that you can put it on a separate partition, but most of the guides were vague on what I needed to do. Could someone give me step-by-step instructions on what I need to do, or even tell me a, easier or different way to fix this /home folder capacity problem? What is the difference between a population and a sample? What is the difference between a population and a sample? What common variables and statistics are used for each one, and how do those relate to each other? Why does my wildcard SSL certificate cause a domain mismatch error on a second level subdomain? Principle of Reflection on atomic level This well-observed phenomenon has, besides several others, always been a fascination to me. We are well aware of several theories, experiments, and practical applications of this well-known phenomenon, but is it established what goes on at the grassroots of reflection?  My high school teacher once told me that whenever light is incident on any reflecting surface, its electrons absorb the energy of the photon and release back the same energy. If this is actually true, then I have plenty of questions to continue, but if it is not, then what is reflection, and how and why does it take place? Should the network use thousands separators in numbers? How can I delete an email address from Outlook's email history? How do I become a Werewolf? I've heard that one of (maybe the only?) secret races in this game is a werewolf.  How do I become a werewolf?  And by extension, how do I cure lycanthropy, that is, if there even is a cure? How to remove illegal characters from path and filenames? If $$S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}\,$$ find the value of $14S$.  The question can be simplified to:  Find $S=\sum\limits_{k=1}^n\,t_k$ if $t_n=\dfrac{n}{1+n^2+n^4}$.  As $1+n^2+n^4$ forms a GP, $$t_n=\frac{n(n^2-1)}{n^6-1}\,.$$ But I can't figure out how to solve further. It would be great if someone could help. How to bottle your own homemade salad dressing How to select (highlight) text in Skype chat without using a mouse? Recently I started learning to use keyboard shortcuts almost for everything I'm doing on the laptop, but soon enough after I started, I bumped into the problem of being unable to highlight (select) simple text in Skype chats.  I did some research and I couldn't find any solution to my problem. Have I missed an already answered question that is the same like mine or if not what solution can you suggest in order to achieve the results I want without touching mouse or touchpad or any cursor moving device? Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to other group I'm currently working on a proof, but didn't get the right idea, a hint would be appreciated:  Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to $\mathbb{Z}/n \times \mathbb{Z}/n$  So, lets assume there is an isomorphism between these groups, my first idea to was to construct a contradiction with the homomorphism property. But I don't know how, there should be an element in  $\mathbb{Z}/n \times \mathbb{Z}/n$ which is not in $\mathbb{Z}/n^2$ to reach this kind of contradiction. Of course there is $n \in \mathbb{Z}/n^2$ which is not in the other group but I don't know if this helps.  Thanks Build and strengthen the Stack Exchange community with "crossover questions" between sites I have been working on this exercise for a while now. It's in B.L. van der Waerden's Algebra (Volume I), page $19$. The exercise is as follows:     The order of the symmetric group $S_n$ is $n!=\prod_{1}^{n}\nu$. (Mathematical induction on $n$.)   I don't comprehend how we can logically use induction here. It seems that the first step would be proving $S_1$ has $1!=1$ elements. This is simply justified: There is only one permutation of $1$, the permutation of $1$ to itself.  The next step would be assuming that $S_n$ has order $n!$. Now here is where I get stuck. How do I use this to show that $S_{n+1}$ has order $(n+1)!$?  Here is my attempt: I am thinking this is because all $n!$ permutations of $S_n$ now have a new element to permutate. For example, if we take one single permutation $$ p(1,\dots,n) = \begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; \dots &amp; n\\ 1 &amp; 2 &amp; 3 &amp; \dots &amp; n \end{pmatrix} $$ We now have $n$ modifications of this single permutation by adding the symbol $(n+1)$:  \begin{align} p(1,2,\dots,n,(n+1))&amp;= \begin{pmatrix} 1 &amp; 2 &amp; \dots &amp; n &amp; (n+1)\\ 1 &amp; 2 &amp; \dots &amp; n &amp; (n+1) \end{pmatrix}\\ p(2,1,\dots,n,(n+1))&amp;= \begin{pmatrix} 1 &amp; 2 &amp; \dots &amp; n &amp; (n+1)\\ 2 &amp; 1 &amp; \dots &amp; n &amp; (n+1) \end{pmatrix}\\ \vdots\\ p(n,2,\dots,1,(n+1))&amp;= \begin{pmatrix} 1 &amp; 2 &amp; \dots &amp; n &amp; (n+1)\\ n &amp; 2 &amp; \dots &amp; 1 &amp; (n+1) \end{pmatrix}\\ p((n+1),2,\dots,n,1)&amp;= \begin{pmatrix} 1 &amp; 2 &amp; \dots &amp; n &amp; (n+1)\\ (n+1) &amp; 2 &amp; \dots &amp; n &amp; 1 \end{pmatrix} \end{align}  There are actually $(n+1)$ permutations of that specific form, but we take $p(1,\dots,n)=p(1,\dots,n,(n+1))$ in order to illustrate and prove our original statement. We can make this general equality for all $n!$ permutations: $p(x_1,x_2,\dots,x_n)=p(x_1,x_2,\dots,x_n,x_{n+1})$ where $x_i$ is any symbol of our finite set of $n$ symbols and $x_{n+1}$ is strictly defined as the symbol $(n+1)$.  We can repeat this process for all $n!$ permutations in $S_n$. This gives us $n!n$ permutations. Then, adding in the original $n!$ permutations, we have $n!n+n!=(n+1)n!=(n+1)!$. Consequently, $S_n$ has order $n!$.  How is my reasoning here? Furthermore, is there a more elegant argument? I do not really see my argument here as incorrect, it just seems to lack elegance. My reasoning may well be very incorrect, however. If so, please point it out to me. What is this type of flexible contact strip called? The LCD below is from a cheap digital clock which was (obviously) damaged (broken glass).    The electrical connection between the PCB and the display was realized with this pink strip.    Its flexible...    ...and consists of alternating rows of conductive and insulating material between the pink boarders.    [above image was altered in contrast/brightness to highlight the alternating strips]  What is this kind of connector called? Is it only used for displays?   I am pretty impressed that the connection is reliable just from the very slight pressure between the strip and the PCB. Also, its easily removable from the display. There seems to be some conductive layer on the glass, from where the signal goes into the displays internals. This connection seems to be damaged for the three rightmost signals, where the glass is broken. It seems that $\{e\}, \{e,s\}, \{e,rs\}, \{e,r^2s\},...,\{e,r^{n-1}s\}, \{e,r,r^2,...,r^{n-1}\}, D_n$ are always subgroups of $D_n$. Especially when $n$ is odd, these seem to be the only subgroups. But when n is even, say $n=4$, then there are also $\{e,s,r^2,r^2s\}$ and $\{e,rs,r^2,r^3s\}$. It makes me wonder, is there a general formula/algorithm for finding all subgroups of $D_n$ when $n$ is even? I installed Ubuntu 12.04 a month ago and am using it till now. I failed to notice that all this time there was no sound at all while running Ubuntu, even while playing a game in Wine. The weird thing is that only the startup sound comes when I log in (Indian/African drum tone), then comes the utter silence.  I tested both Digital Output (S/PDIF) and the speakers in the sound settings but can hear nothing.  Any help?