The following identities are true for a matrix $A$. \begin{align} \dim \mathrm{row}\, A &amp;= \mathrm{rank}\,A \\ \dim \mathrm{Im}\, A &amp;= \mathrm{rank}\, A \\ \dim \mathrm{row}\, A &amp;= \dim \mathrm{Im}\, A^\mathrm{T} \end{align} Does this mean that $\dim \mathrm{Im}\, A^\mathrm{T} = \dim \mathrm{Im}\, A = \mathrm{rank}\,A$? How come?
I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at  and I understand the proof, but I don't "get it". Can someone help me out with this ?   I find it hard to wrap my head around the idea of how the column space and the row space is related at a fundamental level.
existence of sequence of polynomial Is there a sequence of polynomials $P_n$ such that   $\displaystyle\lim_{n \to \infty} P_n(z) = \begin{cases} 1, &amp; \text{Im } z &gt; 0\\ 0, &amp; z \text{ is real,} \\ -1, &amp; \text{Im } z &lt; 0 \end{cases}$  I have no clue where to start from. Please provide some hints. Thanks in advance. What happens when a photon hits a beamsplitter? Yesterday I read that we can affect the path and the 'form' (particle or wave) of a photon after the fact ().  Part of what is puzzling me is the beam-splitter. Are the individual photons actually being split into two new photons of lesser energy?  This  implies that you cannot split a photon but it seems that beam splitters do exactly that. Rounding a number to the nearest 5 or 10 or X Given numbers like 499, 73433, 2348 what VBA can I use to round to the nearest 5 or 10? or an arbitrary number?  By 5:   499 -&gt;  500 2348 -&gt; 2350 7343 -&gt; 7345   By 10:   499 -&gt;  500 2348 -&gt; 2350 7343 -&gt; 7340   etc. Which Pokémon can speak with humans? Inspired by Detective Pikachu, I was wondering which Pokémon can "speak" with humans.  I would like to include  Pokémon like Meowth in the anime that speak English/Japanese with their mouths, Pokémon like Mewtwo in the anime/movies that speak telepathically, or other Pokémon we've seen like Lugia/Entei that seem to speak without moving their mouths but still out loud for all to hear (presumably).  The format doesn't matter (manga, anime, video games). And while this is inspired by Detective Pikachu, it's not about it, as the movie is not out yet      And it seems like the human character has the special ability, not the Pikachu Explicit space character? I'm wondering about the best way to do an explicit space character (I don't even know what it's properly called), like the ones you get in the \begin{verbatim*} environment. The closest I've found is  \sqcup   in math mode, but it seems like a bit of a hack to me. Is there any text symbol for what I'm looking for? Will there be data loss if I upgrade? Distributing elements across a list of lists Does "the same number of people" behave as singular or plural? This hurts my eyes to read it and my ears to say it, but the writer stands by          About the same number of people was awarded bachelor's degrees in 2010 as filed for personal bankruptcy (1.6 million).   Is the subject in this sentence singular or plural? How can I convert a string of bytes into an int in python?   Say like this: 'y\xcc\xa6\xbb'  I came up with a clever/stupid way of doing it:  sum(ord(c) &lt;&lt; (i * 8) for i, c in enumerate('y\xcc\xa6\xbb'[::-1]))   I know there has to be something builtin or in the standard library that does this more simply...  This is different from  for which you can use int(xxx, 16), but instead I want to convert a string of actual byte values.  UPDATE:  I kind of like James' answer a little better because it doesn't require importing another module, but Greg's method is faster:  &gt;&gt;&gt; from timeit import Timer &gt;&gt;&gt; Timer('struct.unpack("&lt;L", "y\xcc\xa6\xbb")[0]', 'import struct').timeit() 0.36242198944091797 &gt;&gt;&gt; Timer("int('y\xcc\xa6\xbb'.encode('hex'), 16)").timeit() 1.1432669162750244   My hacky method:  &gt;&gt;&gt; Timer("sum(ord(c) &lt;&lt; (i * 8) for i, c in enumerate('y\xcc\xa6\xbb'[::-1]))").timeit() 2.8819329738616943   FURTHER UPDATE:  Someone asked in comments what's the problem with importing another module. Well, importing a module isn't necessarily cheap, take a look:  &gt;&gt;&gt; Timer("""import struct\nstruct.unpack("&gt;L", "y\xcc\xa6\xbb")[0]""").timeit() 0.98822188377380371   Including the cost of importing the module negates almost all of the advantage that this method has. I believe that this will only include the expense of importing it once for the entire benchmark run; look what happens when I force it to reload every time:  &gt;&gt;&gt; Timer("""reload(struct)\nstruct.unpack("&gt;L", "y\xcc\xa6\xbb")[0]""", 'import struct').timeit() 68.474128007888794   Needless to say, if you're doing a lot of executions of this method per one import than this becomes proportionally less of an issue. It's also probably i/o cost rather than cpu so it may depend on the capacity and load characteristics of the particular machine. How do you deserialize json properties that are reserved words in apex? Is there any way to deserialize JSON into an object using JSON.deserialize if some of the property names in the JSON are reserved words in apex? I want to do something like this:  string jsonString = '{"currency" : "ABC"}'; public class JSONResult {     public string currency; } JSONResult res = (JSONResult) JSON.deserialize(jsonString, JSONResult.class); system.debug(res);   but of course I get an error that the identifier name is reserved. "provide" vs. "provide with" I am wondering if the following sentence is correct:     We add the information their study provides with to our article.   The context is: their study provides with some information. And we add the information to our article.  I want to keep the word "add", and someone told me that "provides with to" sounds wired... I had an little issue when loggin in with Safari on Mac OS X (El Capitan).   When entering the password in Safari, there is a key symbol that lets you choose among any password that is stored in your keychain. This button interferes with the "forgot?" button that Stack Exchange adds and you can't press the Safari button.     Please move this "forgot?" button below the text field, like most other sites implement this. How to log out of a Stack Exchange site? Per the title, where is the logout button?  All the existing questions I can find refer to the previous design of the site, where hovering over your name in the top bar brought up a menu. Can anonymous class implement interface? Is it possible to have an anonymous type implement an interface?  I've got a piece of code that I would like to work, but don't know how to do this.  I've had a couple of answers that either say no, or create a class that implements the interface construct new instances of that. This isn't really ideal, but I'm wondering if there is a mechanism to create a thin dynamic class on top of an interface which would make this simple.  public interface DummyInterface {     string A { get; }     string B { get; } }  public class DummySource {     public string A { get; set; }     public string C { get; set; }     public string D { get; set; } }  public class Test {     public void WillThisWork()     {         var source = new DummySource[0];         var values = from value in source                      select new                      {                          A = value.A,                          B = value.C + "_" + value.D                      };          DoSomethingWithDummyInterface(values);      }      public void DoSomethingWithDummyInterface(IEnumerable&lt;DummyInterface&gt; values)     {         foreach (var value in values)         {             Console.WriteLine("A = '{0}', B = '{1}'", value.A, value.B);         }     } }   I've found an article  that describes one approach. Is this the best way of doing this? On Windows 7, is there a command prompt line that can start cmd as an administrator? On Linux, we can use  sudo bash   to run a shell as the superuser.  On Windows 7, we can use     Start menu -> (in search box, type) cmd -> Right click on search result and choose Run as Administrator   to run a command prompt as an administrator. Is there also a command prompt line that is like Linux's sudo or su so as to start a shell as an administrator? Which Content Management System (CMS)/Wiki should I use? This is a general, community wiki  question to address non-specific "I need a CMS or Wiki that does x, y, and z..." questions.  If your question was closed as a duplicate of this question and you feel that the information provided here does not provide a sufficient answer, please open a discussion on .    I have a list of features that I want for my website's Content Management System (CMS) - where can I find a [free] script that includes all of them? How can I evaluate a C# expression dynamically? I would like to do the equivalent of:  object result = Eval("1 + 3"); string now    = Eval("System.DateTime.Now().ToString()") as string   Following Biri s , I got this snippet (modified to remove obsolete method ICodeCompiler.CreateCompiler():  private object Eval(string sExpression) {     CSharpCodeProvider c = new CSharpCodeProvider();     CompilerParameters cp = new CompilerParameters();      cp.ReferencedAssemblies.Add("system.dll");      cp.CompilerOptions = "/t:library";     cp.GenerateInMemory = true;      StringBuilder sb = new StringBuilder("");     sb.Append("using System;\n");      sb.Append("namespace CSCodeEvaler{ \n");     sb.Append("public class CSCodeEvaler{ \n");     sb.Append("public object EvalCode(){\n");     sb.Append("return " + sExpression + "; \n");     sb.Append("} \n");     sb.Append("} \n");     sb.Append("}\n");      CompilerResults cr = c.CompileAssemblyFromSource(cp, sb.ToString());     if (cr.Errors.Count &gt; 0)     {         throw new InvalidExpressionException(             string.Format("Error ({0}) evaluating: {1}",              cr.Errors[0].ErrorText, sExpression));     }      System.Reflection.Assembly a = cr.CompiledAssembly;     object o = a.CreateInstance("CSCodeEvaler.CSCodeEvaler");      Type t = o.GetType();     MethodInfo mi = t.GetMethod("EvalCode");      object s = mi.Invoke(o, null);     return s;  } Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$ derivative of cost function for Logistic Regression I am going over the lectures on Machine Learning at Coursera.   I am struggling with the following. How can the partial derivative of   $$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}y^{i}\log(h_\theta(x^{i}))+(1-y^{i})\log(1-h_\theta(x^{i}))$$  where $h_{\theta}(x)$ is defined as follows  $$h_{\theta}(x)=g(\theta^{T}x)$$ $$g(z)=\frac{1}{1+e^{-z}}$$  be $$ \frac{\partial}{\partial\theta_{j}}J(\theta) =\sum_{i=1}^{m}(h_\theta(x^{i})-y^i)x_j^i$$  In other words, how would we go about calculating the partial derivative with respect to $\theta$ of the cost function (the logs are natural logarithms):  $$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}y^{i}\log(h_\theta(x^{i}))+(1-y^{i})\log(1-h_\theta(x^{i}))$$ I'm still a newbie with Linux but I'm trying to learn. I have two USBs, one is 4GB and has the ISO install of Ubuntu (I think it's called live, the one with maximum of 4GB of permanent storage) and got a 64gb usb to make a permanent full install on it. I installed it following  tutorial and everything looks fine (I had to create the EFI partition with the Terminal though) but it just won't boot, the live install boots without problems.   I'm using the school's computers which have the UEFI menu password protected so I can't access that and I won't have a laptop for like 3 weeks (main reason why I'm doing this). I tested it with a friend's laptop disabling the Secure boot but it still wouldn't boot. I also saw that some tutorials don't add the EFI partition so I tried without that one and still won't work. And I know it's not the 64GB usb not booting because I had a live install on it too before and it booted fine.  Please, I have been trying to make it work for two days now, any ideas? I can barely understand this stuff so if you could explain step by step that would be awesome.