Suppose we want to compute the covariance between $z_1$ and $z_2$ given as follows: $$ z_1 = a_1 \cdot x_1 + a_2\cdot x_2 \\ z_2 = a_3 \cdot x_1 + a_4 \cdot x_2\\ % x_1,x_2 \sim p(x) $$ If we compute the covariance between $z_1$ and $z_2$  we have: $$ \mathbb{E}[z_1\cdot z_2] - \mathbb{E}[z_1]\cdot \mathbb{E}[z_2]   = \\ \mathbb{E}[a_1a_3 x_1x_1 + a_1a_4 x_1x_2 + a_2a_3 x_1x_2 + a_2a_4 x_2x_2] - a_1a_3\mathbb{E}[x_1]^2 -a_1a_4\mathbb{E}[x_1]\mathbb{E}[x_2] -a_3a_2\mathbb{E}[x_1]\mathbb{E}[x_2] -a_2a_4\mathbb{E}[x_2]^2 =\\ a_1a_3\text{cov}[x] -a_2a_4 \text{cov}[x] $$ In what I am studying, the last equality holds (or at least that is what I have understood), from the fact that $x_1$ and $x_2$ are independent hence their covariances is zero.
Let $A$ and $B$ be two constant matrices and let $x$ and $ y$ be two random vectors, what is the general formula for $Var(Ax+By)$? I know the formula for when $x$ and $y$ are scalar random variables and $A$ and $B$ are constants, but what about the matrix case?
What is the meaning of "build off of"? What is the meaning of "build off of"? I Googled it, but can't find any definition!  I encountered it in the sentence in a technical book. The sentence is:     For the application we will build, the todos example is the closest   fit, so we'll go ahead and build off of that example.   There is a link related to it I found by Google, But it didn't give the definition! C# difference between == and Equals() Can I visit Chicago during a 5.5 hour layover in O'Hare? Which are the standard commands available in every Linux based distribution? I want to view the fonts that are installed on my system. Specifically, I want the client-side (fontconfig) fonts, not the old-style X11 bitmap fonts on the server (for which xfontsel does an adequate job). I guess what I'm looking for would be called a font viewer or font gallery. I want to see a bit of sample text for each font, ideally configurable sample text.  Being able to filter by font name, available character sets and other characteristics would be a plus.  A search of packages in Debian wheezy found gnome-font-viewer and kfontview, but they can only display a single font. I'd rather avoid depending on software that isn't widely available, including on the current Debian stable and Ubuntu LTS. This question was phrased as a bug report for LaTeX and I thought it would be helpful to be presented here (including an answer).  The person had a lot of "common words" in typewriter font sprinkled around his text and he was surprised that this text would not automatically hyphenate and he had to manually insert \- to mark hypenation points. What should be in the sources.list for Kali Rolling Linux? This is exact content from my sources.list:  #  # deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free #deb cdrom:[Debian GNU/Linux 2016.1 _Kali-rolling_ - Official Snapshot amd64 LIVE/INSTALL Binary 20160830-11:29]/ kali-rolling contrib main non-free   I couldn't install the Synaptic package manager and was told to modify sources.list.  I googled a lot and added some repos into it and ran update and upgrade command only to see Kali crash... I had to recover it.  Please mention the exact content that should be there to ensure that Kali Linux works. What is taking up so much space on my disk, beside the filesystem? Two functions $f:A\rightarrow B$ and $g:B\rightarrow A$. Can someone give me an example where $g\circ f=id_A$ but $f\circ g\neq id_B$? I'm trying to downgrade my iPhone from the current iOS to an older one, as per .  But every time I try, I always receive this error:     The iPhone could not be restored. This device isn't eligible for the requested build.     I have tried many times with and without DFU mode. Any ideas regarding this? How do you create a dropdownlist from an enum in ASP.NET MVC? I'm trying to use the Html.DropDownList extension method but can't figure out how to use it with an enumeration.  Let's say I have an enumeration like this:  public enum ItemTypes {     Movie = 1,     Game = 2,     Book = 3 }   How do I go about creating a dropdown with these values using the Html.DropDownList extension method?  Or is my best bet to simply create a for loop and create the Html elements manually? How to properly hook up various maps types in Cycles? I created some texture maps in MindTex to use in Blender, but I can't figure out how to properly hook them up. The maps I have:  - Diffuse  - Normal  - Height   - Specular   - Gloss   - Self Illumination  - Reflection  - Ambient Occlusion   MindTex (left) vs. Blender (right)  As you can clearly see it's a pretty big difference.  is my material node setup in Blender. In the examples I don't actually use self illumination or reflection images, but including this in the answer would be helpful. Can you be refused a visitor visa to US if you have bad Credit there? I did my 3 years of College in the US, 2010-2013 due to finance issues I was unable to pay my credit card bill of a certain bank, which still lie unpaid.  However I plan on visiting the US later this year; could I be denied visa for having an unpaid bill ? When Deadpool is in the hallway talking      he makes a joke about Fox springing for a few more X-Men and the camera pans to show Beast closing the door to get rid of the noise.   Who else was in the room? Parsing input with scanf in C I've been having a lot of problems trying to figure out how to use scanf().  It seems to work fine with integers, being fairly straight forward scanf("%d", &amp;i).  Where I am running into issues is using scanf() in loops trying to read input.  For example:  do {   printf("counter: %d: ", counter);   scanf("%c %c%d", &amp;command, &amp;prefix, &amp;input); } while (command != 'q');    When I enter in a validly structured input like c P101, it seems to loop again before prompting me.  This seems to happen even with a single:  scanf("%c", &amp;c)    in a while loop.  It'll do the loop twice before prompting me again.  What is making it loop twice, and how do I stop it? When I enter in less amount of input that programmatically wouldn't have another character or number such as q, pressing enter seems to prompt me to enter more.  How do I get scanf() to process both single and double character entries? $f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $ Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$ and $f'(x) = [f(x)]^{2}$, $\forall x \in \mathbb{R}$. Show that $f(x) = 0$,  $\forall x \in \mathbb{R} $.  I first (unsuccessfully) tried using the Mean Value Theorem, but this is in the Integrals chapter so the solution probably involves them. Can't really see where integrals come in here though.  What I've got so far:   (i) Since $f$ is differentiable, thus it is continuous and, hence,  integrable. Therefore $f^2$ is also integrable and as $f'=f^2$, $f'$ is too.   (ii) $f' \geq 0$, $\forall x \in \mathbb{R}$ Getting inverse of a transposition key Determining the Orbits/Orbit Space of $O(3)$ on Real 3 by 3 Traceless Symmetric Matrices Let $O(n)$ be the set of $3 \times 3$ orthogonal matrices, and $S$ be the set of traceless real symmetric $3 \times 3$ matrices.  Now  $O(n)$ will act on $S$ by conjugation. Note that by the spectral theorem $ \forall A \in S$ in the orbit of $A$ there is a diagonal matrix $D=diag(\lambda_1, \lambda_2,\lambda_3)$ with $\lambda_1\le\lambda_2 \le\lambda_3 $.  Determine the all the orbits of the action &amp; the orbit space of the action.  What I've done:  Since all elements of $S$ are diagonlizable that the orbit of any element should be the orbit of its' diagonalized form because $\forall A \in S,   \exists P \in O(3)$ st  $PAP^t=D $ with $D$ a diagonal matrix and moreover for any other  $Q\in O(3)$ we get $QDQ^t=QPAP^tQ^t=(QP)A(QP)^t=RAR^t$.   And so would that imply the orbit space is just the set of traceless diagonal matrices as $tr(QDQ^t)=tr(DQ^tQ)=tr(Q)$ (cyclic property of trace) and $Q\in S$.  Any help is appreciated. How can I install Broadcom Wireless Adapter BCM4352 802.11ac PCID [14e4:43b1] (rev 03) on fresh install of Ubuntu 14.10 (Utopic Unicorn)? I have a brand new Dell XPS 15. I   dual boot it with Windows 8.1 and  Ubuntu 14.10  The Wireless Network Adapter is Broadcom BCM4352 802.11ac with PCID [14e4:43b1] (rev 03).  The 4352 is not included in the list for installation via bcmwl-kernal-source file. I tried this anyway, but the Terminal command led to the file starting to install, but then stopping. Sometimes I get a permission denied error message.  As a result of reading and trying all advice from forums I have decided to completely reinstall Ubuntu 14.10. I doubt my particular Broadcom Wireless Card has no support on Ubuntu 14.10?    Edit:  The advice of Chili555 worked. I now have working wifi using Broadcom Wireless Adapter BCM4352 802.11ac PCID [14e4:43b1] (rev 03) How to install pip for python 3 in ubuntu 12.04 LTS