Determining the matrix of a linear transformation Let $A$ be an $n\times n$ matrix, and let $V$ denote the space of $n$-dimensional row vectors. What is the matrix of the linear operator ‘‘right multiplication by $A$’’ with respect to the standard basis of $V$?
Matrix of Linear Transformation by right multiplication I am trying to solve the following problem:   Let $A$ be an $n\times n$ matrix, and let $V$ denote the space of $n$-dimensional row vectors. What is the matrix of the linear operator ‘‘right multiplication by $A$’’ with respect to the standard basis of $V$?  I am not sure where to begin with this problem.
I have a British passport from 2006 September and going to Florida. The passport in has camera symbol. Is it valid? I am going to Florida in a few days. I have a British passport that I acquired in September 2006. It has a "camera" symbol on front and is biometric. Just seen news items that only passports from October 2006 or later will be accepted by the USA. Is my passport valid for travel to the USA? I do have a valid ESTA. No transaction fee ETF trades - what's the catch? I just noticed TD Ameritrade has program where you can trade from a list of ETFs with zero fees.  Sounds like it could save me a bit of money, but my cynical side wonders how they make money off such a program...  Is it a loss leader? (They entice me to have an account there, and then make money off the other trades I make?)  Or do they simply want to grow the size of their asset base -- what is the mechanism by which they make money on the funds that I have in my account?  Edit: in case it wasn't clear in the original question above, this is an ongoing program for free ETF transactions -- not a one-time deal for some number of free trades when you open an account. I have had an account there for years and I'm eligible to participate -- it's not a program explicitly designed just to gain new accounts. Calculating square roots using the recurrence $x_{n+1} = \frac12 \left(x_n + \frac2{x_n}\right)$ Let $x_1 = 2$, and define $$x_{n+1} = \frac{1}{2} \left(x_n + \frac{2}{x_n}\right).$$ Show that $x_n^2$ is always greater than or equal to $2$, and then use this to prove that $x_n − x_{n+1} ≥ 0$. Conclude that $\lim x_n = \sqrt2$.   My question:  So I know how to do this problem but I don't know how to prove that $x_n &gt;0$ for all $n\in \mathbb{N}$. I’m a big fan of ampersands (possibly too much …) and I’d like to replace all mentions of “et al.” (included in citations) by \&amp; al. – both in the main text (I use natbib citations of the form [Author1 et al., 1999]), and in the actual bibliography?  How can I do that, considering that BibTeX is inserting the words for me, and I have no control over the text? Preferably, I’d like to do this without creating my own bibliography style sheet (since I want to make this work regardless of the actual BibTeX style that is used), but if anyone can tell me how I need to edit the style sheet file, that would be fine as well (makebst doesn’t seem to offer an option for this …). TZInfo::DataSourceNotFound error starting Rails v4.1.0 server on Windows Is there an easier way to install Chromium than what is described ?  Furthermore, is there an easier way to stay up to date with current Chromium releases rather than having to re-build Chromium each release? (a script which automatically grabs and installs the latest build possibly?).  Related: What happens to Bucky's arm at the end of Infinity War? Bucky (Winter Soldier) has a metal arm (possibly made from vibranium?) but it's an attachment to his body as far as I understand. Why, when Thanos destroys half of the universe at the end of Infinity War, does his metallic arm also disappear with him? Shouldn't it be left behind as his gun did? Prove that $$\cos(\pi/11)+\cos(3\pi/11)+\cos(5\pi/11)+\cos(7\pi/11)+\cos(9\pi/11)=1/2$$ using Euler's formula.  Everything I tried has failed so far.  Here is one thing I tried, but obviously didn't work. $$\Re e \{e^{\frac{\pi}{11}i}(1+e^{\frac{2\pi}{11}i}+e^{\frac{4\pi}{11}i}+e^{\frac{6\pi}{11}i}+e^{\frac{8\pi}{11}i}) \}=\frac{1}{2}$$ $$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{e^{2\pi i}}+\sqrt[11]{e^{4\pi i}}+\sqrt[11]{e^{6\pi i}}+\sqrt[11]{e^{8\pi i}}) \}=\frac{1}{2}$$ $$\Re e \{e^{\frac{\pi}{11}i}(1+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}+\sqrt[11]{1}) \}=\frac{1}{2}$$ $$\Re e \{5e^{\frac{\pi}{11}i} \}=\frac{1}{2}$$ $$5\cos(\frac{\pi}{11})=\frac{1}{2}$$ Which isn't true :D  Thanks in advance What canon evidence is there that Stormtroopers are/are not still clones? Reading this article about , my nerd-rage started flaring as they claimed that Stormtroopers were still Fett-Clones 20 years after the rise of the Empire.  As I started thinking about it, I realized that everything I had read saying that Modern-Day Stormtroopers not being clones was from the EU - C Level books or games.   One of the quotes often cited for backing the idea that the Stormtroopers are clones comes from Princess Leia:     Aren't you a little short for a Stormtrooper?   But through the movie, paying close attention, Stormtroopers are often shown to be slightly different heights. One even hits his head in the doorway while others walk through uninjured.                                  Is there any G Canon fully canon (non-Legends/Infinities) evidence to support the idea that Stormtroopers are, or are not still clones at the time of Episode IV? How can I create and extract zip archives from the command line? Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance! PHP __PHP_Incomplete_Class Object with my $_SESSION data How can all labels of a given type of node be restyled in Tikz?  Imagine that there are nodes of different styles, and that only the nodes of style X should have red labels. Is this possible without having to change all labels one by one of nodes of style X? How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$  (a) has a fixed point and that (b) has a point $x$, such that $f$ maps $x$ to its antipodal point $-x$? How to set up USB for Virtualbox? I'm using Ubuntu Maverick and Virtual Box 3+. I have Windows 7 Ultimate. The problem is that it doesn't detect USB drives in Windows 7, but USB peripherals (mouse + keybord) are working.  I followed , but I can't find the user group vboxusers. How can I continue? I would also like to upgrade to the latest version without any risk (without losing my virtual hard disk drive). How do you set, clear, and toggle a single bit? "Could not identify document class." error when trying to count words I'm trying to count words in a cyrillic LaTeX document with .  I run the command perl texcount.pl index.tex and get following output:  D:\wd&gt;perl texcount.pl index.tex  !!! Could not identify document class. !!! File: index.tex Encoding: utf8 Words in text: 0 Words in headers: 5 Words outside text (captions, etc.): 0 Number of headers: 1 Number of floats/tables/figures: 0 Number of math inlines: 0 Number of math displayed: 0  (errors:1)   index.tex starts with following line:  \documentclass[a5paper,pagesize,10pt,bibtotoc,pointlessnumbers, normalheadings,DIV=9,twoside=false]{scrbook}   So documentclass tag is present.  How can I fix this error? When I was at school and learning integration in maths class at A Level my teacher wrote things like this on the board.  $$\int f(x)\, dx$$  When he came to explain the meaning of the $dx$, he told us "think of it as a full stop".  For whatever reason I did not raise my hand and question him about it.  But I have always shaken my head at such a poor explanation for putting a $dx$ at the end of integration equations such as these.  To this day I do not know the purpose of the $dx$.  Can someone explain this to me without resorting to grammatical metaphors? When do the differences between APS-C and full frame sensors matter, and why? I found this interesting problem in :     Given a point $P$ in space, and given a piece of malleable material of constant density, how should you shape and place the material in order to create the largest possible gravitational field at $P$?   Any ideas?