I have a matrix A, described using an environment bmatrix as in the MWE below.    \documentclass{article}  \begin{document} Consider the matrix  $A=\begin{bmatrix}  1 &amp; 0 &amp; -1 \\  2 &amp; 3 &amp; 1  \\   \end{bmatrix}$  Then its transpose is  $A^T=\begin{bmatrix}  1 &amp; 2 \\ 0 &amp; 3\\  -1 &amp; 1  \\   \end{bmatrix}$  \end{document}   How can I get the transpose of A without typing all its entries? In other word, is there an environment where \ means "end of column" instead of "end of row"?  I hope I could just paste and copy the definition of A and then change bmatrix to vbmatrix or whatever.
As writing matrices in LaTeX is an extremely time-consuming activity, I was wondering if a software or a site exists that automatically gave the transposed form of a matrix, that is to say, given a matrix:  \begin{bmatrix}    0 &amp; -1 &amp; -1 \\    -1 &amp; -1 &amp; 0 \\    -1 &amp; 0 &amp; 1 \\    1 &amp; 0 &amp; 0 \\    0 &amp; 0 &amp; -1 \\    -1 &amp; 2 &amp; 1  \end{bmatrix}   it will produce a transposed matrix, with the columns in place of the lines and conversely...  It would be particularly time-saving!
Hi I'm solving some exercise problems in my text : "A Course in Mathematical Statistics".  I'm in the chapter "Point estimation" now, and I want to find a UMVUE of $\theta$ where $X_1 ,...,X_n$ are i.i.d random variables with the p.d.f $f(x; \theta)=\theta e^{-\theta x}, x\gt0$.   I know that $E(X_i)=1/\theta,$ for each $i$, and also have that $\bar{X}$ (or equivalently $\sum_1^n X_i$) is a complete sufficient statistic for $\theta$. But I cannot go any further here. Somebody can help me? I'm trying to compile a document that only works with an up-to-date distribution, so I'd like to know: how do I update my distribution? Please provide screenshots where appropriate.  List of answers           To keep an eye on what’s going on at CTAN you may subscribe to the  or the RSS feed feed://www.mail-archive.com/ctan-ann@dante.de/maillist.xml Geometric mean never exceeds arithmetic mean This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.)     The geometric mean of $n$ positive numbers $a_1, a_2,\ldots,a_n$ is $\sqrt[n]{a_1a_2 \ldots a_n}$ and their arithmetic mean is $\frac{a_1+a_2+\ldots+a_n}{n}$. If $a_1, a_2,\ldots,a_n$ are $n$ positive real numbers, prove by induction that their geometric mean is always smaller than or equal to their arithmetic mean, i.e. $\sqrt[n]{a_1a_2\ldots a_n} \leq \frac{a_1+a_2+\ldots+a_n}{n}$ Why does the symbol for the multiplication operation change shape? Find a circle's radius with three known tangent lines I need to find the equation for a circle (mainly its radius) which is tangent to the following three lines:  $y = 0$  $y = \tan(70)x$  $y = -1.428148x + 0.790201$  For the last tangent line equation, I know that it is tangent at the point $(0.371272, 0.259968)$ However, for the other two I do not know the exact point of tangency, only that the circle is tangent to them. Asymptotic expansion for Fresnel Integrals If you take the fresnel integrals to be $$S(x) = \int_{0}^{x}\sin \left(\frac { \pi \cdot  t^2}{2} \right) dt$$ How do you find the asymptotic expansion? I know it begins with a $1/2$ but how? Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$ "It was my op" what does it mean? I heard this in  movie. It was my op. What does it mean? Googling gave me : 'Original Poster'. But that doesn't seem very meaningful. Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$, $\forall x\in G$. Prove that $H$ is a normal subgroup of $G$.    I have tried to using the definition but failed. Can someone help me please. Spam iCloud Calendar Invitations What is the best way to generate a GUID/UUID from Apex code (such as a trigger)? Preferably in the following format:     nnnnnnnn-nnnn-nnnn-nnnnnnnnnnnnnnnnn   Example:     13219ec0-3a81-44c5-a300-de14b7d0235f Where does the idiom "beating around the bush" come from? Where does the idiom "beating around the bush" come from? I recently purchased a MacBook Pro 13&quot; with the M1 chip and transferred all my data over using a Time Machine backup. Homebrew packages are currently installed in /usr/local/opt/ but it is my understanding that they should be under /opt/homebrew for Apple Silicon Macs. How can I:  Get the list of packages I currently have installed Remove those from /user/local/opt Reinstall them under /opt/homebrew  I kind of assumed Homebrew would be clever enough to do this automatically but apparently it needs some manual maintenance. Here's my Homebrew config if it helps: HOMEBREW_VERSION: 2.7.2 ORIGIN: https://github.com/Homebrew/brew HEAD: dad7dc6a1498b80770d98f2d7cd6fb927c300bbb Last commit: 3 days ago Core tap ORIGIN: https://github.com/Homebrew/homebrew-core Core tap HEAD: 2b1d79d038bffdbfcee93283051d48919a4caf3d Core tap last commit: 16 hours ago Core tap branch: master HOMEBREW_PREFIX: /usr/local HOMEBREW_REPOSITORY: /usr/local/Homebrew HOMEBREW_CELLAR: /usr/local/Cellar HOMEBREW_CASK_OPTS: [] HOMEBREW_MAKE_JOBS: 8 Homebrew Ruby: 2.6.3 =&gt; /System/Library/Frameworks/Ruby.framework/Versions/2.6/usr/bin/ruby CPU: octa-core 64-bit arm_firestorm_icestorm Clang: 12.0 build 1200 Git: 2.30.0 =&gt; /usr/local/bin/git Curl: 7.64.1 =&gt; /usr/bin/curl macOS: 11.1-arm64 CLT: 12.3.0.0.1.1607026830 Xcode: N/A Rosetta 2: false  Thanks in advance! How to clear the Skype chat history how clear the skype chat history ? Should I use the past tense with did? For example, I was to say:     The important question is: Did they knew what it means or not?   Or should I say:     The important question is: Did they know what it means or not?   In other words: Should I use past tense with did?  I looked at this:    but it didn't really answer my question. Right identity and Right inverse implies a group Let $(G, *)$ be a semi-group. Suppose   $ \exists e \in G$ such that $\forall a \in G,\  ae = a$;  $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$.    How can we prove that $(G,*)$ is a group? For $n \geq 2$, prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$ For $n \geq 2$ prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  We need to use induction.   The Principle of Mathematical Induction, Theorem 4.2.1, states that $n_0 \in \mathbb{Z}$. For each integer $n \geq n_o$, let $P(n)$ be a statement about $n$. Suppose that the following two statements are true:   $P(n_o)$ $( \forall n \geq n_o)[P(n) \rightarrow P(n+1)]$   Then, for all integers $n \geq n_o$, that statement $P(n)$ is true.  Suppose,  $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$, the equation, is true for some natural number $k$.  $\frac{1}{2^2})*...*(1- \frac{1}{k^2}) = \frac{k+1}{2k}$  The induction hypothesis $P(k) \rightarrow P(k+1)$ must hold for any natural number $k, P(k),$ from the basis and inductive step.  $(\frac{1}{2^2})*...*(1- \frac{1}{k^2})*(1- \frac{1}{(k+1)^2}) = \frac{k+1}{2k}(1- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $\frac{k+1}{2k}(\frac{(k+1)^2}{(k+1)^2}- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $(\frac{(k+1)}{2k})( \frac{(k+1)^2-1}{(k+1)^2})$  $\leftrightarrow$ $( \frac{(k^2+2k+1-1}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{(k^2+2k}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{k+2}{(2)(k+1)})$  We need to demonstrate that   $(\frac{1}{2^2})*...*(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  holds true for $n \geq 2$  The question is do I just substitute a number that is greater than two after I did the induction? Grub not loading after Windows 8 Install My system was configured to dual boot Ubuntu 12.04.1 LTS and Windows 7. Today I got my hands on the MSDN release of Windows 8 and I installed it over my Windows 7. Now the computer just boots to Windows 8 directly without loading the GRUB screen.  So I followed the steps as suggested in: . Running this command:      ls -l /dev/disk/by-label/   gives the following output:      total 0     lrwxrwxrwx 1 root root 10 Sep 11 07:51 Entertainment -&gt; ../../sda2     lrwxrwxrwx 1 root root 10 Sep 11 02:45 PENDRIVE -&gt; ../../sdb1   Also fdisk -l command gives this as the output:      Disk /dev/sda: 320.1 GB, 320072933376 bytes     255 heads, 63 sectors/track, 38913 cylinders, total 625142448 sectors     Units = sectors of 1 * 512 = 512 bytes     Sector size (logical/physical): 512 bytes / 512 bytes     I/O size (minimum/optimal): 512 bytes / 512 bytes     Disk identifier: 0x1246aa23         Device Boot      Start         End      Blocks   Id  System     /dev/sda1   *        2048   319582199   159790076    7  HPFS/NTFS/exFAT     /dev/sda2       319582208   602906623   141662208    7  HPFS/NTFS/exFAT     /dev/sda3       602908672   625135615    11113472   83  Linux      Disk /dev/sdb: 1939 MB, 1939865600 bytes     64 heads, 63 sectors/track, 939 cylinders, total 3788800 sectors     Units = sectors of 1 * 512 = 512 bytes     Sector size (logical/physical): 512 bytes / 512 bytes     I/O size (minimum/optimal): 512 bytes / 512 bytes     Disk identifier: 0xc3072e18         Device Boot      Start         End      Blocks   Id  System     /dev/sdb1   *        2248     3788799     1893276    c  W95 FAT32 (LBA)   So I assume that I have to run this: sudo grub-install /dev/sda3 to get GRUB up and running. But I am getting this error:      /usr/sbin/grub-probe: error: cannot find a device for /boot/grub (is /dev mounted?).   Can anyone please guide me in the right direction? The current Ubuntu installation is far too customized to my needs to lose it to a boot manager issue! Any help is much appreciated! Currently I had built a file upload functionality in Lightning component which accepts the CSV file, read it as text(Each row in CSV is split using \n and each column in the row is split using ,). Then each columns in each row are validated in Javascript and if everything is fine,the data is saved in the backend. While uploading the data, if the file contains , in its data then it is taken as another column. Eg: X-36936,SS-341487,11-WT,414081,"LENDERS, THE HACK",2-Manage,,,,,,,,,,,,12/1/2013,3150,,0,,Dealing,2018,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,  Here "LENDERS, THE HACK" is a single column but it is taken as 2 columns (LENDERS --> 1st column and THE HACK --> 2nd column).Could anyone please help me how to resolve this issue? How do you do a deep copy of an object in .NET? I want a true deep copy. In Java, this was easy, but how do you do it in C#?