Prove that if $T:V \rightarrow U$ is an invertible linear transformation then the inverse $T^{-1}: U \rightarrow V$ is also linear Prove that if $T:V \rightarrow U$ is an invertible linear transformation then the inverse $T^{-1}: U \rightarrow V$ is also linear So, $T$ is linear and thus $T(v_1+v_2) = T(v_1)+T(v_2)$ for all $v_1,v_2 \in V$. Also, $T$ is invertible and so we can write $T^{-1}T(v)=v$ for all $v \in V$. I'm a bit confused trying to put this all together, help appreciated!!
Is the inverse of a linear transformation linear as well? A question from the field of Linear Algebra. If I have a linear transformation $T$ that is one-to-one and onto, would that mean that the $T^{-1}$ will also be linear? If so, is there any general proof for it?  Thanks
EDIT:  My question was poorly worded.   I wasn't asking about showing $\ln(1+x) &gt; \frac{2x}{2+x}$ for $x&gt;0$.   What I wanted to know is why the lower bound provided by $ \frac{2x}{2+x}$ was so tight for small positive values of $x$. This is addressed in robjohn's answer. Should I install programs from a source tarball (`.tar.gz`), from the Ubuntu Software Centre, or from elsewhere? Pretty formattings for warning and info blocks I'm looking for pretty formatting for warnings, and others for some info or tips.  For the moment, I use the following formatting but maybe someone has one better idea.    To obtain this, I use the following code.  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[x11names,svgnames]{xcolor} \usepackage{graphicx} \usepackage{fourier-orns} \usepackage[framemethod=tikz]{mdframed} \usetikzlibrary{calc}  \makeatletter % Default settings for warnings     \newcommand{\warningSymbol}{\raisebox{0.8\depth}{\danger}}      \definecolor{warningColorText}{named}{Red3}     \definecolor{warningColorLine}{named}{Red3}     \definecolor{warningColorBack}{named}{LemonChiffon1}     \definecolor{warningColorBackSymbol}{named}{white}  % Inline mode     \newcommand{\warning}[1]{%         \textcolor{warningColorText}{\warningSymbol{}\,#1}%     }  % Block mode     \tikzset{         warningsymbol/.style={             rectangle,             draw  = warningColorText,             fill  = warningColorBackSymbol,             scale = 1,             overlay         }     }      \mdfdefinestyle{warning}{%         hidealllines      = true,         leftline          = true,         skipabove         = 12,         skipbelow         = 12pt,         innertopmargin    = 0.4em,%         innerbottommargin = 0.4em,%         innerrightmargin  = 0.7em,%         rightmargin       = 0.85em,%         innerleftmargin   = 1.1em,%         leftmargin        = 0.85em,%         middlelinewidth   = .2em,%         linecolor         = warningColorLine,%         backgroundcolor   = warningColorBack,%         fontcolor         = warningColorText,%         firstextra        = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)             node[warningsymbol] {\warningSymbol};         },%         secondextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         middleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         singleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%     }      \newmdenv[style=warning]{@Warning}     \newenvironment{Warning}{\let\warning\relax\begin{@Warning}}{\end{@Warning}} \makeatother  \usepackage{lipsum}  \begin{document}  \warning{IMPORTANT !} \lipsum[1]  \begin{Warning}      \lipsum[1] \end{Warning}  \end{document} Proof that $a\mid b \land b\mid c \Rightarrow a\mid c $ I am trying to prove:     $a\mid b \land b\mid c \Rightarrow a\mid c $   $a\mid b$ means that a divides b if there is an integer k, that $b=k\cdot a$  Please give me a hint on how to start, because I have no idea. Touch enabled monitors and Android Gray cone obscuring/clipping the 3D Viewport Add AD users to sudo group not working | CentOS 6.7 How do I get a tourist visa (or a tourist card) for Cuba?  I live in the Netherlands, but I'm not a Dutch citizen, I have an Italian passport. Avoiding "(Not Responding)" label in windows while processing lots of data in one lump I occasionally need to process a large amount of data from one package off the network, which takes sufficiently long that when the user tries to interact with the application windows adds the "(Not Responding)" string to the window title. I am aware this is because the processing is being done within a call to handle a message (some way up the stack) and therefore is blocking the message pump. I'm also aware the ideal way to deal with this is to process the data asynchronously in a separate thread so the pump can continue running, however this is a LARGE desktop application which is single threaded from top to toe and safely spinning this processing off is not feasible in our time frame.  So with that in mind, is there by any chance a way I can at least avoid the "not responding" moniker (which to most users reads as "has crashed") by telling windows my application is about to be busy before I begin the work? I believe there is something along these lines when responding to a request to close, one can keep asking windows for more time to avoid it proclaiming that your not "closing in a timely fashion"  I should add this is a C++ MFC application. mathtools: \dddot cause the term to rise How can a piece of A4 paper be folded in exactly three equal parts? This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on eye estimate where to fold. Then I turn the letter over and fold bottom to top. Most of the time ending up with three different areas. There must be a way to do this exactly, without using any tools (ruler, etc.). Is there any module or some configuration that I can define the meta tags for the views page display in drupal 7 using  3.0? The question of  made me wonder as to whether wizards, any of them, do practice religion. Whether there be magic-person only religions or shared muggle faiths, I do not recall any wizards (born in the wizard world anyway) saying that they did practice a religion.  Many of them do celebrate Christmas, but as was commented on the above linked question, there are many non-christian muggles that also observe Christmas. So I am not sure that can be an accurate identifier. Answering this question may link into identifying . Compare the statistical significance of the difference between two polynomial regressions in R So first of all I did some research on this forum, and I know   questions have been asked but they usually haven't been answered properly or sometimes the answer are simply not detailed enough for me to understand. So this time my question is : I have two sets of data, on each, I do a polynomial regression like so :  Ratio&lt;-(mydata2[,c(2)]) Time_in_days&lt;-(mydata2[,c(1)]) fit3IRC &lt;- lm( Ratio~(poly(Time_in_days,2)) )   The polynomial regressions plots are:    The coefficients are :  &gt; as.vector(coef(fit3CN)) [1] -0.9751726 -4.0876782  0.6860041 &gt; as.vector(coef(fit3IRC)) [1] -1.1446297 -5.4449486  0.5883757    And now I want to know, if there is a way to use an R function to do a test that would tell me whether or not there is a statistical significance in the difference between the two polynomials regression knowing that the relevant interval of days is [1,100].  From what I understood I can not apply directly the anova test because the values come from two different sets of data nor the AIC, which is used to compare model/true data.  I tried to follow the instructions given by @Roland in the related question but I probably misunderstood something when looking at my results :  Here is what I did :   I combined both my datasets into one.  f is the variable factor that @Roland talked about. I put 1s for the first set and 0s for the other one.  y&lt;-(mydata2[,c(2)]) x&lt;-(mydata2[,c(1)]) f&lt;-(mydata2[,c(3)])  plot(x,y, xlim=c(1,nrow(mydata2)),type='p')  fit3ANOVA &lt;- lm( y~(poly(x,2)) )  fit3ANOVACN &lt;- lm( y~f*(poly(x,2)) )   My data looks like this now :    The red one is fit3ANOVA which is still working but I have a problem with the blue one fit3ANOVACN the model has weird results. I don't know if the fit model is correct, I do not understand what @Roland meant exactly.  Considering @DeltaIV solution I suppose that in that case :  The models are significantly different even though they overlap. Am I right to assume so ? Word for feeling conflicting emotions simultaneously Detecting unused packages in LaTeX document I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? As far as I can tell, it's something like this:  damage reduction = armor / (armor + 100)   Is this accurate? Why is the edit button disabled? The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$  of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail method. In case more solutions are existing or not how to check? What are the solutions of $x^4 - 2y^2 = 1$? Debian 10 systemd, how to disable autostart of graphical environments I installed a graphical environment on a server and I have decided that I no longer want to run that environment at boot. How can I disable it with systemd? This is gnome desktop environment.