I want to prove $\exp : \mathfrak{gl}(n, \mathbb{C}) \longrightarrow GL(n,\mathbb{C})$ is surjective.   The textbook gives hint as  "using Jordan canonical form"  So my guess is expressing matrix in terms of similarity transformation, but how does it guarantee that the exponential map is surjective?   Can you give me some proof of above statement.
Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by  $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for all $A\in M_n(\mathbb{C}).$  I want to prove that $\exp$ is a surjective map from $M_n(\mathbb{C})$ to $GL(n,\mathbb{C})=\left\{A\in M_n(\mathbb{C})\,\middle| \det(A)\neq0\right\}$, how do I go about that?   I mean saying that $\exp:M_n(\mathbb{R})\to GL(n,\mathbb{R})$ is an analogous to saying $\exp:\mathbb{R}\to \mathbb{R}_{&gt;0}$ and this is also pretty intuitive, since, in analogy with the case of numbers, $A^0=I\;\forall A$, so $\exp(0)=I+0+\frac{0^2}{2!}+\dots=I$, so even for $A=0$ we get $\det\left(\exp(A)\right)\neq0$ and so because of the first term we can never get a zero determinant. But I have no idea how to prove the subjectiveness.  Thanks in advance.
Can you nest html forms? Is it possible to nest html forms like this  &lt;form name="mainForm"&gt;   &lt;form name="subForm"&gt;   &lt;/form&gt; &lt;/form&gt;   so that both forms work? My friend is having problems with this, a part of the subForm works, while another part of it does not. With Ubuntu 16.04 laptop doesn't suspend when lid is closed I have a Dell 7737 laptop, and with Ubuntu 16.04 it no longer suspends when the laptop lid is closed. With 14.04 closing the lid put the laptop to suspend as expected. However, I reloaded the system with Ubuntu 16.04 and now closing the laptop does nothing. The laptop continues to run as if nothing happened.  When I manually select "Suspend" from the menu it works as expected. (Though it does prompt me to double login on resume. No, not encryption login followed by OS login. This is a double OS login.)  NOTE: Altering the System Settings|Power doesn't seem to affect anything.  Any ideas on what changed from 14.04 to 16.04 in this area? Is the electric force a vector or a vector field? Accelerating electrons via microwaves In Synchrotrons I think they use microwaves to accelerate the electrons bundles that fly through-how does putting a microwave through a cavity accelerate an electron? I know that the Electric and Magnetic fields could and do have an effect on the electron, but if the radiation is moving parallel to the electron, wouldn't the force be perpendicular? Also, how exactly does the microwave radiation look in the vacuum chamber? Are the two ends constant nodes? This is a really weird situation. A guy contacted me regarding a business I have. He offered far in excess of my stated fees and even told me he could help me financially if I needed it, which I do. He said he was a doctor. Before he even showed up, he gave me his bank account and routing number and full name, which I used to pay some utility bills. All the payments went through. He also sent me $2500 through PayPal, and I received an email that I had received the payment, though it's been 24 hours and it still hasn't shown up in my "recent activity" in my account. But at the time when I met him, I had no reason to doubt the whole thing was real.   So we met, and through conversation he seemed totally legit. Halfway through our session, he abruptly left — he claimed he was being called in to the hospital. He had already received some, but not all, of the services we had agreed to. He said he would come back in a few hours. Oh yeah, and since the PayPal funds hadn't gone through yet, and I needed cash to pay rent, he said his "financial advisor" would be wiring me $800 a couple hours later. Well, I never heard back from him and never got a call about a wire transfer. In retrospect it all seems very obviously a scam, except for the fact that he gave me a bank account number that actually worked as far as paying some bills!  My question is, assuming the PayPal payment never goes through — I never got an email saying that it was cancelled, but it isn't showing up in my payments either. I'm planning on calling PayPal if it hasn't gone through by tomorrow. But assuming that payment doesn't go through, how can I use the bank account info to get the payment he promised me as far as the services I provided? Other than paying all my utilities in advance? You'd think with an account number and routing number and full name, I would be able to withdraw money somehow, but I really have no idea how. And since it isn't under my name, I can't make a payment on my student loans. I wouldn't take more than he owed me, but it would really be nice to at least get that, even if I didn't get the $2500 he had promised me.   You always hear about not letting people see your account number and stuff like that, but it seems impossible to do anything with this info even though I have it! If scamming people with their bank account number is so easy, how do scammers do it? So I'm wondering two things:   What legal way can I take what I am owed from this guy? What illegal way do people use this info if they had it? I don't want to get in trouble, but I'm just curious because you always hear how easy it is.   (Oh, and I never actually saw his ID so I don't know that the name &amp; bank info he gave me was actually his, he could have stolen it from someone, though he really didn't seem like a hacker — he was a middle aged black man who was very smart-sounding, polite, soft spoken, and he seemed to have a lot of knowledge that a real doctor would have. I have a college degree and am very skeptical in general and careful about not being scammed and I've never had an experience even close to this, and the whole time I kept thinking "is this guy for real?" but since the utility stuff had gone through, it seemed legit. And nothing he did sent up a red flag like he might be a criminal or something. But later I googled the name and couldn't find a doctor by that name, at least not in the state he said he lives in. And since he said he would be contacting me last night, and he never did, it seems obvious that he planned the whole thing, but it just makes no sense to me.) When to use virtual destructors? SharePoint Migration 2007 to 2013 I"m studying normalization and have been reading through descriptions and examples online. One Concept that isn't clear to me is if the process can be automated at any point. I understand that a system can't tell you if you schema makes any real-world sense, but it seems likely to me that if you feed a program some arguments say, the functional dependencies, it would be able to do some steps for you.  Say you have a relationship R(A,B,C,D)  and you tell your program that A,B->C and A->D and that the primary key is (A,B)  Now if my understanding serves me correctly it should be possible for a program to identify that A->D violates 2NF because D depends on A and A is a subset of the primary key. That the proper decomposition would be R1(A,B,C) and R2(A,D).   The main book i'm learning from and my instructor indicate these normalization's are done all by hand. Is this correct or can some parts be automated? Who is responsible for defining a site logo? I mean, there are moderators of the site that define it, or stackexchange defines it? I'm trying to find the derivative of $\sin^{-1}(x)$. I know the steps that lead to $\frac{1}{\sqrt{1-x^2}}$, however I don't understand why the following reasoning leads to a wrong answer.  Because  $$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} $$  If we plug in for $f(x) = \sin(x)$, and because $\frac{d}{dx}\sin(x) = \cos(x)$ we get  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\sin^{-1}(x))} $$  Since $\sin(x) = \cos(x-\frac{\pi}{2})$, we can state that $\sin^{-1}(x) = \cos^{-1}(x-\frac{\pi}{2})$. (I suspect this is what is wrong)  Thus,  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\cos^{-1}(x-\frac{\pi}{2}))}$$  Then, by the definition of inverse function, we have  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{x - \frac{\pi}{2}}$$ Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$      Attemption $$ \begin{cases} f(-1) = -a - b - 16\\ f(2) = 8a + 2b - 28 \end{cases} $$ But then when I plug in $$b = 28 - 8a$$. Then I get $$-a -(28- 8a) - 16=0$$$$a = 34/7$$  In the book it says $a = 10$ and $b = -26$. So I am stumped. Please help Why Does the Transaction Log Keep Growing or Run Out of Space? This one seems to be a common question in most forums and all over the web, it is asked here in many formats that typically sound like this:  In SQL Server -  What are some reasons the transaction log grows so large? Why is my log file so big? What are some ways to prevent this problem from occurring? What do I do when I get myself on track with the underlying cause and want to put my transaction log file to a healthy size? Puppet client not connection to master - SSL connect Error I installed the puppet master and agent on the same machine. When the client is started, I got the following error message.  puppet agent --server=agent.com --no-daemonize --debug    err: Could not retrieve catalog from remote server: SSL_connect returned=1 errno=0 state=SSLv3 read server certificate B: certificate verify failed.  This is often because the time is out of sync on the server or client I am simply adding user in 'Monitored Users' in Debug Logs. But User is not getting added. We all know that a simple statement such as:  SELECT * FROM stuff;   should not produce an ordered result. However, when I try to demonstrate this, it always comes out in primary key order.  Moreover a statement such as:  SELECT thing,whatever FROM stuff GROUP BY thing,whatever;   always appears to sort thing by the last field in the GROUP BY clause, which is not at all helpful.  The question is, under what circumstances does SQL SERVER order the result even if unasked, and what can I do to stop this?  I am trying to demonstrate to my students that order is indeterminate unless specified, but this is not helping my case.  I admit that I am working with a small set of sample data.  Thanks Measuring the distance between lines and points in QGIS Using QGIS - I have a vector of fault lines.  I have point data in the form of a csv which I have also loaded into QGIS.  I need to know how far each point is from the nearest fault line then enter this distance in a new column in the csv file. Any ideas? Title is removed/cleared/erased when starting editing Possible Duplicate:          As answered in  I want to use git diff --color=always filename | less -r to get a colored output of my git diffs. Since I use this a lot it would be nice to set up an alias for this. But how to include the filename in this command?  I tried the following without success:  alias gdiff='git diff --color=always $1 | less -r'   Typing gdiff example.php displays the content of example.php with less but ignores the diff part.  Any ideas? I have six upvotes for the one question with no intervening history. They have been summarised into two groups - one with 4 events and one with 2 events. However, when you expand either, both show all 6 events. Minor I know, just letting you know. I have a Windows Server 2012 R2 Domain.  Yesterday, a computer’s (running Windows 10 Pro) network drive stopped working.  After further investigation (gpresult /h) it appears ALL group policy objects are failing with the reason Inaccessible, Empty, or Disabled.  I have confirmed that all the GPOs still exist and are enabled on both (redundant and local) domain controllers.  Furthermore, there are 20 other machines on the same domain and LAN with absolutely no problems.  However, there is one other computer that I tested which presented with the same problem!  Does that mean the problem is with the servers?  gpresult /r reports that one client is getting GPOs from local DC1, and the other from DC2.  So it is not a problem related to a specific DC.  gpupdate /force fixed nothing (though it claimed that policies were applied).   I tried deleting the registry entries for local policies (following this guide ) and rebooting - same problem.    I found this support page from Microsoft (), but it claims it only applies to Windows 7 or earlier clients.  All my machines (both server and client) are running 64-bit versions and are fully updated.  I have rebooted all of them just to be sure.