How to prove that if $f^3 = f$, $f$ is diagonalizable?
Proving that if $A$ is a $8\times 8$ matrix over $\mathbb{R}$ and $A^3=A$, then $A$ is diagonalizable.
Redirect console output to textbox in separate program My child has chickenpox, am I infectious? My 3-year-old has had chickenpox for about 2-3 days, but a close friend has just had a baby and I want to go and help. Is it possible that I am infectious, despite not having the illness? I had it as a child, so (hopefully) can't actually get it myself. Thanks! Why is a conservative force defined as the negative gradient of a potential? I'm learning about work in my dynamics class right now. We have defined the work on a particle due to the force field from point A to point B as the curve Integral over the force field from point A to B. From math I know that if a vector field has a potential, we only need to evaluate the potential at point B minus the potential at point A to get the result of the curve Integral. In the text that I'm reading, it's explained that if the integral over a force-field is path-independent, then the force field $F = -{\rm grad}(V)$, where $V$ is the potential. Why is it defined as the negative gradient? Doesn't one determine the potential from $F$ mathematically. Why do we impose the sign on the potential? Windows Starter Edition, Home and Home Premium do not include gpedit, how do I install it? Why are some actors uncredited in certain films? Eli Wallach  in Mystic River. Olivia Williams  in X-Men: The Last Stand. However, neither of them (according to IMDb) is credited in the movie. These two instances are simply examples.  Why are some actors, who are oftentimes quite well known, not credited in certain features? Why is $\frac{\operatorname dy'}{\operatorname dy}$ zero, since $y'$ depends on $y$? I know that $\frac{dy'(x)}{dy}=0$ (where $y'=\frac{dy(x)}{dx}$). The reason explained is that $y'$  does not depend explicitly on $y$. But intuitively, $y'$ depends on $y$, since if you vary $y$ you will modify $y'$. Why is my reasoning wrong (my reasoning sounds like it's related to functional calculus, instead of standard calculus)?  I tried to write  $\frac{dy'(x)}{dy}=\frac{d}{dy}\frac{dy}{dx}=\frac{d}{dx}\frac{dy}{dy}=\frac{d}{dx}1=0$, but this proof doesn't convince me.  I think that other way to see this is saying that if a function is of the form $f(y)$, it will not dependend on the variable $y'$.  But the same way you write $f(y)=y^2$, you could write $f(y)=\frac{d}{dx}y$, which clearly depends on $y'$. So I don't know if there are some types of operations which are restricted (for example, taking limits):  Note: The problem raises in the context of Classical Mechanics, where: $\frac{\partial L(\dot{x},t)}{\partial x}=0$. All rounds of AES (and Rijndael) have a MixColumns step, save the last round which omits it.  DES has a similar feature where the last round differs slightly.  The rationale, if I recall correctly, was to "make the cipher appear similar in reverse as it does in the forward direction."  Why does this make AES$^{-1}$ look similar to AES?  And how does this help in implementing the block cipher? Why do video cables (DVI, VGA) still screw in? Differentiability implies continuity - A question about the proof I have a question, to aid my understanding, about the proof that differentiability implies continutity.$\mathstrut$    Differentiability Definition  When we say a function is differentiable at $x_0$, we mean that the limit:  $$‎f^{\prime} ‎(x) = \lim_{x\to x_0} \frac{f(x) - f(x_0)}{x-x_0}$$ exists.  Continuity Definition  When we say a function is continuous at $x_0$, we mean that: $$\lim_{x\to x_0} f(x) - f(x_0) = 0$$  Theorem: Differentiability implies Continuity: If $f$ is a differentiable function at $x_0$, then it is continuous at $x_0$.  Proof:  Let us suppose that $f$ is differentiable at $x_0$. Then $$ \lim_{x\to x_0} \frac{f(x) - f(x_0)}{x-x_0} =  ‎f^{\prime} ‎(x) $$  and hence  $$ \lim_{x\to x_0} f(x) - f(x_0) = \lim_{x\to x_0} \left[ \frac{f(x) - f(x_0)}{x-x_0} \right] \cdot \lim_{x\to x_0} (x-x_0) = 0$$  We have therefore shown that, using the definition of continuous, if the function is differentiable at $x_0$, it must also be continuous.  My Question  The proof seems to execute the following steps:   Assume the function is continuous at $x_0$ Show that, with little algebra, we can change this into an equivalent question about differentiability at $x_0$. With this little bit of algebra, we can show that if a function is differentiable at $x_0$ it is also continuous.   What I am slightly unsure about is the apparent circularity. In my mind it seems to say, if a function is continuous, we can show that if it is also differentiable, then it is continuous. Rather than what I was expecting, namely, if a function is differentiable, we can show it must be continuous.  Hopefully my confusion is clear. Any help will be greatly appreciated. I want create my own IDE but I want a code editor that would syntax highlighting.  I want to embed it as a control into a WPF window.  Malcolm What is the correct possessive for nouns ending in "‑s"? What is the possessive of a noun ending in ‑s?  Are these both right, or is the second one wrong?   the boys' books the boss' car What is the difference between fastboot oem unlock and fastboot oem unlock-go? I'm having issues unlocking my Nexus 7 2013 WiFi's bootloader, and it was suggested that I try fastboot oem unlock-go instead of fastboot oem unlock. Both seem to act the same on my device, however both also fail on my device so it is possible that the difference appears after the failure on my end comes. Regardless, I would like to know: What is the difference between these two commands? I'm not managing to find documentation. I just saw a user named: .yahoo.co.jpaqwsykcj3aulh3h1k0cy6nzs3isj  Could new users be named according to their user-number, or something other than "Unknown"? Perhaps User1234567 instead? It would be less awkward to address them in answers and comments.   Additionally, I often times have to check profiles to make sure users aren't abusing the system, since I find many of the same questions from "unknown (google)" from time to time. That, or I'll mistakenly think I've helped the user on a similar issue in the past, and assume the user has knowledge that he or she does not have. I have a ListBox DataTemplate in WPF. I want one item to be tight against the left side of the ListBox and another item to be tight against the right side, but I can't figure out how to do this.  So far I have a Grid with three columns, the left and right ones have content and the center is a placeholder with it's width set to "*". Where am I going wrong?  Here is the code:  &lt;DataTemplate x:Key="SmallCustomerListItem"&gt;     &lt;Grid HorizontalAlignment="Stretch"&gt;         &lt;Grid.RowDefinitions&gt;             &lt;RowDefinition/&gt;         &lt;/Grid.RowDefinitions&gt;         &lt;Grid.ColumnDefinitions&gt;             &lt;ColumnDefinition/&gt;             &lt;ColumnDefinition Width="*"/&gt;             &lt;ColumnDefinition/&gt;         &lt;/Grid.ColumnDefinitions&gt;         &lt;WrapPanel HorizontalAlignment="Stretch" Margin="0"&gt;             &lt;!--Some content here--&gt;             &lt;TextBlock Text="{Binding Path=LastName}" TextWrapping="Wrap" FontSize="24"/&gt;             &lt;TextBlock Text=", " TextWrapping="Wrap" FontSize="24"/&gt;             &lt;TextBlock Text="{Binding Path=FirstName}" TextWrapping="Wrap" FontSize="24"/&gt;          &lt;/WrapPanel&gt;         &lt;ListBox ItemsSource="{Binding Path=PhoneNumbers}" Grid.Column="2" d:DesignWidth="100" d:DesignHeight="50"      Margin="8,0" Background="Transparent" BorderBrush="Transparent" IsHitTestVisible="False" HorizontalAlignment="Stretch"/&gt;     &lt;/Grid&gt; &lt;/DataTemplate&gt; What decides what to prepend the duplicate question list with on a duplicate question? On duplicate questions, I often see different headings for the duplicate question list. These include:     Possible Duplicate:    [duplicate question list]   and      This question already has an answer here:    [duplicate question list]   and more recently,     This question is an exact duplicate of:    [duplicate question list]   What decides which heading is placed before the duplicate question list? What if residuals are normally distributed, but y is not? I've got a weird question. Assume that you have a small sample where the dependent variable that you're going to analyze with a simple linear model is highly left skewed. Thus you assume that $u$ is not normally distributed, because this would result in normally distributed $y$. But when you compute the QQ-Normal plot there is evidence, that the residuals are normally distributed. Thus anyone can assume that the error term is normally distributed, although $y$ is not. So what does it mean, when the error term seems to be normally distributed, but $y$ does not? How to vertically align an image inside a div Applying MVT and IVT (?) to show something about f'''? (edit: actually, Taylor's Theorem using Lagrange remainder) What Nikon lens for both macro and portrait? All three lenses are available at a local store near me:   85mm 1.8 at about 400USD  85mm 1.4 at about 1000USD 105mm 2.8 at about 1015 USD   I plan on doing macro and portrait photography and have the ff thoughts so far:   I can save a lot with the 85mm 1.8 and a site claims that AF-wise it is much faster than the 85mm 1.4 at twice the price 105mm has VR but I don't know if that would be of much use with portrait and macro as I usually don't use slow shutter speeds with such subjects 105mm has AF-S which means I can get better insect shots because of the silence( and the 105 mm range) I don't know why I should prefer the 85mm 1.4 over the 105mm, when the 85mm may only have AF speed as an advantage, though I am not so sure if it is faster   so as you can see, I'm leaning towards 105mm f/2.8 or 85mm f/1.8 and less towards the 85mm f/1.4. However, I don't want to finalize the decision as many also claims that the 85mm f/1.4 is so good.  Any professional insights to add in helping me make the decision? Is a Schengen visa/transit visa required for a transit flight through two Schengen airports?