So I have this matrix  $$     \begin{pmatrix}     -2 &amp; 0 &amp; 0 \\     3 &amp; 1 &amp; -6 \\     0 &amp; 0 &amp; -2 \\     \end{pmatrix} $$  Finding Nul(A+2Id) gives me  $$     \begin{pmatrix}     0 &amp; 0 &amp; 0 \\     3 &amp; 3 &amp; -6 \\     0 &amp; 0 &amp; 0 \\     \end{pmatrix} $$  On the answers sheet it says to solve the homogeneous system associated with A + 2Id which gives us   Nul(A+2Id)=Span $$     \begin{pmatrix}     1 &amp; -1  \\     1 &amp; 1  \\     1 &amp; 0  \\     \end{pmatrix} $$  I am completely lost on how to find the Span Nul(A+2Id)=Span $$     \begin{pmatrix}     1 &amp; -1  \\     1 &amp; 1  \\     1 &amp; 0  \\     \end{pmatrix} $$  Can somebody please help me with the steps in order to get to this solution ?  Moreover, on the answers sheet is states  Since the dimension of the eigenspace E(λ1) is equal to 2 we observe that the ma- trix A is diagonalizable.  How do i know from this solution that the eigenspace E(λ1) is equal to 2?  Thanks!
So i have this $3\times 3$ matrix  $$     \begin{pmatrix}     -2 &amp; 0 &amp; 0 \\     3 &amp; 1 &amp; -6 \\     0 &amp; 0 &amp; -2 \\     \end{pmatrix} $$  I want to check if the matrix is diagonizable. First thing i do is find the roots : namely $\lambda_1 = −2$ of multiplicity $2,$ and $\lambda_2 = 1$ of multiplicity $1.$  In order to ensure that the matrix is diagonalizable we need to find a basis of $\mathbb{R}^3$ whose elements are eigenvectors for $A.$ In particular, since we have the eigenvalue $\lambda_1=-2$ having multiplicity $2$ we need to check that the dimension of the eigenspace $E(\lambda_1)$ is equal to $2.$ We compute the eigenspaces of $A.$  $\DeclareMathOperator\Nul{Nul}$On the answer sheet is says that we need to find $\Nul(A+2Id)$... but why $2Id?$ Isn't is supposed to be $-2,$ so $\Nul(-2Id-A)$??   I do not understand this $\Nul(A+2Id).$ I always thought that we should compute $\Nul(-2Id-A),$ so what do i need to do ?   $\Nul(-2Id-A)$ gives me a different result from $\Nul(A+2Id).$  Does that mean that since $-2$ has a multiplicity of $2,$ we need to work with $2$ and not $-2$?  Somebody please help with a clear and simple explanation ! Thanks!
Error installing Apache2 Server Renaming user name I made a mistake in the username at the time of installation. So I changed the name of the user after logging in by creating a guest account with SUDO rights. But when I checked the list of users using ls /home command I am not seeing the new name for the user instead Old name is appearing. But I use the new name to login.   Is it like I need to change the name inside Home directory separately? Also , if there is any other changes needs to be done .. ? How to fix "-5 No address associated with hostname error" while updating? Stack exchange badges It might be useful to encourage existing users of stackoverflow to cross pollinate other stack exchange sites.  While rep-sharing is not useful, as rep is meant to be a measure of expertise in one subject, badges may be one way to encourage participation across many sites.  This was triggered by seeing how few people actually had greater than 1k rep on all three main sites (SO, SF, SU) via Dennis Williamson's  app.  I had tried to do this by hand prior to the API under the assumption that many people would be answering questions on all three sites, but it turns out there are only 33 people, of all the users of stackoverflow, that have a 1k+ reputation on all three sites.  Over time it may be useful to provide badges that appear on all sites one is participating in that are awarded for extraordinary participation in the whole universe of Stack Exchange sites.  For instance, a silver badge might appear on all my profiles if I obtain 1k rep on 5 or more sites, and a gold badge if I have more than 3k rep on 5 or more sites.  I would hesitate extending this much past 5 sites as we don't want to encourage spamming questions and answers on sites where the user has little interest, but across the universe of stackapps I know I'm proficient or expert in at least 10 current sites, and while it's not a big nudge, the idea that I can get extra awards - especially ones that appear on all sites I'm in - would move me to participate a little more on sites that I am registered on, but don't visit regularly.    I expect, eventually as sites are launched, most users of any one site will have some interest or knowledge in at least 5 other sites.  This may provide the extra nudge some sites need to maintain an excellent rating in terms of usage. Prove that ${x dy - y d x\over x^2 + y^2}$ is not exact Over the last 3-4 weeks I have been trying to find a rogue DHCP server on my network but have been stumped! It is offering IP Addresses that do not work with my network, so any device that needs a Dynamic Address is getting one from the Rogue DHCP and then that device stops working. I need help to find and destroy this thing! I think it might be a Trojan of some sort.  My Main Router is the only valid DHCP Server and is 192.168.0.1 which offers a range of 192.160.0.150-199, and I have this configured in my AD as Authorized. This ROGUE DHCP claims to be coming from 192.168.0.20 and offering an IP Address in the range of 10.255.255.* which is messing up EVERYTHING on my network unless I assign a static IP Address to it. 192.168.0.20 does not exist on my network.  My network is a single AD Server on Windows 2008R2, 3 other physical servers (1-2008R2 and 2 2012R2) about 4 Hypervisor VM's, 3 laptops and a Windows 7 box.  I can't ping the rogue 192.160.0.20 IP, and I can't see it in the ARP -A output, so I can't get its MAC address. I'm hoping that someone reading this post has come across this before. How to install TeX on Android phones? I used  app in my Android phone. But this app does not support XeLaTeX and requires an internet connection to be used.  Is there any distribution of TeX for Android? How I can install TeX on my Android phone? resetPassword problem for enterprise users when called from a community? I'm working within a custom application in a customer community that is used by both community-license users and Salesforce enterprise users.  There is a custom Visualforce page where an application admin can select a user and reset the password.  The VF Controller is using the standard System.resetPassword() method.  We are finding that when an enterprise user receives this password reset email, the user can't get to the expected password reset page.  Instead, it eventually redirects to the community login screen.  The password reset URL for these enterprise users is based on the community's URL, and it seems that these users cannot change password through the community -- it has to be through the enterprise URL, like what you get when you reset password via Setup.  Has anyone run into this and found a workaround?  (Of course community users have no trouble with password resets through this Visualforce page.) How to make a field NULL using MySQL GUI Tools? I'm using the MySQL Query Browser (part of the ) and need to change a field to NULL, but I can't figure out how to do it - if I delete the value it tries to update it to ''. Typing "NULL" makes it try to update to 'NULL' (a string).  I know I could just write a query to do it, but that defeats the entire purpose of the tool, no? \sum caused \sqrt to not be slanted Consider the question :    The large number of down-votes displays as 199 rather than a score of -199.  This is clearly unintentional (and unfortunate for questions that possess such voting!), but can this be&nbsp;corrected? I am just curious, I use handlebar mounted lights, and since I have multiple bikes to switch, it's a lot of trouble changing the light from one to another. So, I am thinking of switching to helmet mounted lights.  Give me your point of view. Why or why not? Which is better suitable for which riding type? What's a way to implement a flexible buff/debuff system? Overview:  Lots of games with RPG-like statistics allow for character "buffs", ranging from simple "Deal 25% extra damage" to more complicated things like "Deal 15 damage back to attackers when hit."  The specifics of each type of buff aren't really relevant.  I'm looking for a (presumably object-oriented) way to handle arbitrary buffs.  Details:  In my particular case, I have multiple characters in a turn-based battle environment, so I envisioned buffs being tied to events like "OnTurnStart", "OnReceiveDamage", etc.  Perhaps each buff is a subclass of a main Buff abstract class, where only the relevant events are overloaded.  Then each character could have a vector of buffs currently applied.  Does this solution make sense?  I can certainly see dozens of event types being necessary, it feels like making a new subclass for each buff is overkill, and it doesn't seem to allow for any buff "interactions".  That is, if I wanted to implement a cap on damage boosts so that even if you had 10 different buffs which all give 25% extra damage, you only do 100% extra instead of 250% extra.  And there's more complicated situations that ideally I could control.  I'm sure everyone can come up with examples of how more sophisticated buffs can potentially interact with each other in a way that as a game developer I may not want.  As a relatively inexperienced C++ programmer (I generally have used C in embedded systems), I feel like my solution is simplistic and probably doesn't take full advantage of the object-oriented language.  Thoughts?  Has anyone here designed a fairly robust buff system before?  Edit: Regarding Answer(s):  I selected an answer primarily based on good detail and a solid answer to the question I asked, but reading the responses gave me some more insight.  Perhaps unsurprisingly, the different systems or tweaked systems seem to apply better to certain situations.  What system works best for my game will depend on the types, variance, and number of buffs I intend to be able to apply.  For a game like Diablo 3 (mentioned below), where nearly any bit of equipment can change a buff's strength, the buffs are just character stats system seems like a good idea whenever possible.  For the turn-based situation I'm in, the event-based approach may be more suitable.  In any case, I'm still hoping someone comes along with a fancy "OO" magic bullet which will allow for me to apply a +2 move distance per turn buff, a deal 50% of damage taken back to the attacker buff, and a automatically teleport to a nearby tile when attacked from 3 or more tiles away buff in a single system without turning a +5 strength buff into its own subclass.  I think the closest thing is the answer I marked, but the floor is still open.  Thanks to everyone for the input. Is the 64-Bit version of Ubuntu only compatible with AMD CPUs? I was told that computers with more than 2 gig's memory need a 64 bit operating system to utilize all RAM.  Is the 64bit Ubuntu download really JUST for AMD processors? I am asking because the disk image I downloaded says AMD64.  So will my new Intel 2.3Ghz Core i3 Dual Core processor work with 64 bit Ubuntu?  It runs the 64bit version of Windows without any qualm. Cycles : shadows on smooth shaded mesh, strange behavior I wonder if this is a bug, or if I missed something... When a light hits tangentially convex surfaces with "smooth shading", the shadow on this surface seems to "partially ignore" the smoothness of the surface. Here is an example with a simple cylinder :    On the left, a cylinder with "flat shading". Everything's ok. On the right, the same with "smooth". We can see an edge, bad shadowed.  With or without the "Edge split" modifier, no matters, the edge is still there. The shader is a simple white diffuse shader with no texture nor displace.  I have to add a subsurf modifier with at least 2 levels, which will insanely increase the vertex count... And I can't do that for my project.  I hope someone already saw it, and have a little tip to share. Or if someone already know it's a bug, or a limit of the current Cycles renderer ? TestMethod do not support Web service callouts, test skipped Choosing coordinate system to use in South Africa? What are the principal datums and projections used in South Africa including the names of geodetic datums and their projections? I made a map with annotations in QGIS 3.0.  This is how it looks like in the layout view:    If I print the map as PDF, one would expect to get the PDF to be WYSIWYG, right? However, it looks like this (it's about half the size):    Is there a way to influence the the scaling of the Annoations? I already tried to play with the Scale and Magnifier in the map view. I also tried to change the Scale in the map item.   The strange thing is, I believe it worked a few days ago. But today I installed QGIS 3.0 on a new PC and it showed the same behaviour... Book series about time travel from future to distant past, psychic powers? Why are papers without code but with results accepted? I just started reading some papers (Computer Science, specifically Computer Vision) and thought "now let's look at the source code" and was quite astonished that most of the papers don't have any source code implementing the described methods to look at, while claiming some performance or being better than other papers.  How do these papers get accepted in journals / conferences? Do people have to submit their source code privately to the reviewers at least, so that they can reproduce the experiment if possible.  Do most journals / conferences just "trust" that people who submit the paper really implemented the theory and got those exact results?  I always had this idea that any experiment should be reproducible by others else it's not scientific justified. Been wondering about this the last few days.