Let $\{(a_{ij},...,a_{in}) \in T^n \mid i = 1,...,s \le n\}$ be a set of vectors. Prove that if $$|a_{jj}| \gt \sum_{i=1, i\neq j}^s|a_{ij}|, \quad 1 \leq j \leq s$$ then the set is linearly independent.  My attempt: I was trying an example with $n=s=3$,  $|a_{11}| \gt $$\sum_{i=1, i\neq j}^{n=3}|a_{i1}|= |a_{21}|+|a_{31}|$   $|a_{22}| \gt $$\sum_{i=1, i\neq j}^{n=3}|a_{i1}|= |a_{12}|+|a_{32}|$  $|a_{33}| \gt $$\sum_{i=1, i\neq j}^{n=3}|a_{i1}|= |a_{13}|+|a_{23}|$  For the linear independence we would have to satisfy, that $x=y=z=0$  $$         \begin{pmatrix}         a_{11} &amp; a_{12} &amp; a_{13} \\         a_{21} &amp; a_{22} &amp; a_{23} \\         a_{31} &amp; a_{32} &amp; a_{33} \\         \end{pmatrix}           \begin{pmatrix}         x \\         y \\         z \\         \end{pmatrix} = \begin{pmatrix}         0 \\         0 \\         0 \\         \end{pmatrix} $$  But I just don't know what's next. Like the $|a_{jj}|$ are on the places where are pivots and if these elements are after Gauss elimination nonzero, then our given set in linearly independent. But I don't know how to prove it. Thanks for help.
Let $a_1,\dots ,a_n \in \mathbb{R}^n$ with $a_i = (a_{i1}, \dots , a_{in})$.  Show that when $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid \; \forall \; j\in \{1,\dots ,n\}$, then $a_1,\dots ,a_n$ are linearly independent.  I want to show that the matrix \begin{matrix} a_{11} &amp; a_{21} &amp; \cdots &amp; a_{n1} \\ a_{12} &amp; a_{22} &amp; \cdots &amp; a_{n2} \\ \vdots &amp; &amp; \ddots &amp; \\ a_{1n} &amp; a_{2n} &amp; \cdots &amp; a_{nn} \end{matrix} can be transformed into an upper triangular matrix with all entries in the main diagonal $\neq 0$.  For the cases $n=1$ and $n=2$ this is easy. However, I fail to show this for the general case; the problem being that the tranformations mess up the coefficients such that I can't use the assumption on the $a_{jj}$'s anymore.  I'm pretty sure that this has already been solved somewhere on here, but I don't know how to search for it.
Using notify-send with cron I'm trying to do something similar to this epsilon-delta graph I've found.  Specifically, I'd like to copy the placements and arrows, like the L+e and L-e coming out of L. Same thing with the x0 + d and x0-d coming out of the x0. Can someone help me with this? This is what I have so far.    \begin{tikzpicture} \begin{axis}[ ,         legend pos=north east,         axis x line=middle,         axis y line=middle,         x tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         y tick label style={/pgf/number format/fixed,                             /pgf/number format/fixed zerofill,                             /pgf/number format/precision=1},         grid = major,         width=8cm,           height=6cm,         grid style={dashed, gray!30},         xmin=0   ,     % start the diagram at this x-coordinate         xmax= 4,    % end   the diagram at this x-coordinate         ymin= 0,     % start the diagram at this y-coordinate         ymax= 11,   % end   the diagram at this y-coordinate %axis background/.style={fill=white},         xlabel=$x$,         ylabel=$f(x)$,         tick align=outside,         enlargelimits=false] % plot the stirling-formulae %\addplot[domain=0:200, red, ultra thick,samples=100] {K/(1 +    ((K-X0)/X0) * exp(r*x))    }; \addplot[domain=0:200, red, ultra thick,samples=100] {3*x-1 };  \addplot[line width=1pt, dashed, mark=none, black] coordinates {(0,8) (3,8)}; \addplot[line width=1pt, dashed, mark=none, black] coordinates {(3,0) (3,8)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(3.33,0) (3.333,9)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (2.666,0)}; \addplot[line width=1pt, dashed, mark=none, blue] coordinates {(2.666,7) (0,7)}; \addplot[line width=1pt, solid, mark=none, blue] coordinates {(3.333,9) (0,9)}; \addplot[holdot] coordinates{(60,8.66)};  % 1\addlegendentry{$f(x) = 10 \sin (x)$}  \end{axis}   \end{tikzpicture}   which gives How do I show apt-get package management history via command line? It seems that Geometric Complexity Theory requires much knowledge of pure maths such as algebraic geometry, representation theory.   While I am a CS student and do NOT have classes of very abstract and pure mathematics, I am interested in this program.  Is there a list of "minimum knowledge" for learning this theory ?     This list includes lecture notes of CS or math departments, surveys of any journal or conferences, and text books of pure math.  [EDIT: Added later] Thank you for your comments.  General theory of computing: I read Sipser's book with the title "Introduction to the Theory of Computation"  Complexity Theory :In particular, I am interested in concrete models for lower bounds of complexity. Thus I read the part of the "concrete lower bounds" in Arora-Barak's text book. I also have basic knowlede in the several chapers of the communication complexity book written by Nisan.  Basic Mathematics : I have learnt about proof-based linear algebra such as general definition of the vector space etc and precicely argument of calculas based on epsilon-delta argument.   Algebra: I have learned about definition and examples of group, ring, and field. I had a class for cs students, and have not learned about general thoery of this algebraic systems. From all the hard drives I have bought, they never seem to be as large as the advertised size; from 320 GB down to 290 GB, from 500 GB down to 450 GB, etc. Is there a technical reason for this? How to open pdf file in Windows Phone 8? How can I create a cross product like the one in the picture.    \documentclass{article} \usepackage{tikz} \begin{document}  \begin{tikzpicture}[line width=.7pt] \draw [-&gt;] (-.2,0)node[below]{$o$}--(4.5,0)node[below]{$x$}; \draw [-&gt;] (0,-.2)--(0,2)node[left]{$y$};  \end{tikzpicture}  \end{document} Clip tool gives ERROR 000732? Evaluating $\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$ Evaluating $$\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$$   I had read the , but it doesn't work. How can I pass command-line arguments to a Perl program? I'm working on a Perl script. How can I pass command line parameters to it?  Example:  script.pl "string1" "string2" Best practices for catching and re-throwing .NET exceptions Manually clear saved question drafts I've seen lately that new question editing is stored somewhere for later use if we wander off the new question without posting it and then return back to the page. Which is a good functionality addition in it's own right.  What I'd like to also have is the Discard this draft functionality that would clear form data and forget it. Somehow we find an answer in the meantime and the question isn't valid any more.  So having a Discard saved draft or something similar would be a nice contra-functionality to auto-save to remove stored data. The way it's implemented now seems too hidden hidden and unpredictable to users.  Additional edit  When adding hidden auto functionality usually means creating the opposite as well by either giving the possibility to preference of not having automatism at all or giving the possibility to clear what's been auto saved.  Let's see how GMail doesn't work (and I think we all agree that from UX perspective it does the job as expected:   User clicks Compose message User starts writing a new email message User decides to not send the message and checks their inbox User clicks Compose message again A clean form is presented to user (old one being part of drafts)   I'm not suggesting we should have something like Saved drafts but we surely should have the possibility to clear previous question draft.  Since this new functionality was introduced I should point out that I would use the discard functionality more frequently than the saved one. I do welcome saving messages, but discarding of saved drafts is anything but a rare situation one might want to do. It wouldn't be a problem if this was a 1% against 99%. To me it's more like 50/50.  Best possible solution  When providing saved drafts there should (from the UX perspective) be the opposite functionality as well. Namely discarding of it. But first thing we should be asking ourselves is:    do users more frequently need pre-filled saved question draft or do users more frequently need empty form with the possibility to populate it with a saved draft   User testing should provide the usage frequency between the two and the one that's more likely to happen (my guess is that there would be a big difference between the two) should be implemented. So one of these:   Pre-populate question fields with saved draft on question form load, while also providing a button to clear the draft and start writing anew without the need to manually clear all the fields (which would be the optimal solution in case of a single field form) Question form should always load empty while also having a button that would pre-populate it with a saved draft.   I think this would be the best consensus related to saved question drafts and their usage. But as we've seen this issue has been swept under the rug for an indefinite time. Make Applications (Ubuntu Software) to access all the DRIVES in my computer Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous. Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty$. Prove that $f$ is uniformly continuous.   Since $f(x)\to 0$ as $x\to \pm\infty$ we have $|f(x)|&lt;\epsilon $ whenever $x&lt;-K$ and $x&gt;K$ where $K&gt;0$.  Also $f$ is uniformly continuous on $[-K,K]$ as the domain is compact.  Hence $f$ is uniformly continuous in $(-\infty,-K)\cup [-K,K]\cup (K,\infty)$.  But how to conclude that $f$ is uniformly continuous on $\Bbb R$ from above as I may chose $x\in (-\infty,-K);y\in [-K,K]$ ,how to conclude that $|f(x)-f(y)|&lt;\epsilon $ from above for a given $\epsilon$?  Please suggest the edits required. How do I disable touchpad when using a mouse Cell identification by Body Cells How do the B-cells and T-cells recognize or distinguish a body cell from a foreign particle?  I suspect something to do with this causes Auto Immune Disorders. In a technical environment, what is the most suitable sentence to use when answering to someone about a problem that they had and we solved it for them:   The problem is solved The problem has been solved Drawing a 3D cylinder, and a circle shape on the x---y plane I'm trying to draw a cylinder on the three dimensional space, and I also want its projection on the x--y plane. I found the first answer  to be very useful. I just copy-paste it to my document, and modify it as follows (the modifications are only inside the \begin{tikzpicture} tags):   \documentclass[parskip]{scrartcl} \usepackage[margin=15mm]{geometry} \usepackage{tikz}  \pgfmathsetmacro{\xdeg}{30} \pgfmathsetmacro{\xx}{cos(\xdeg)} \pgfmathsetmacro{\xy}{sin(\xdeg)}  \pgfmathsetmacro{\ydeg}{120} \pgfmathsetmacro{\yx}{cos(\ydeg)} \pgfmathsetmacro{\yy}{sin(\ydeg)}  \pgfmathsetmacro{\zdeg}{80} \pgfmathsetmacro{\zx}{cos(\zdeg)} \pgfmathsetmacro{\zy}{sin(\zdeg)}  \newcommand{\tdcyl}[5]{% origin x, origin y, origin z, radius, height     \path (1,0,0);     \pgfgetlastxy{\cylxx}{\cylxy}     \path (0,1,0);     \pgfgetlastxy{\cylyx}{\cylyy}     \path (0,0,1);     \pgfgetlastxy{\cylzx}{\cylzy}     \pgfmathsetmacro{\cylt}{(\cylzy * \cylyx - \cylzx * \cylyy)/ (\cylzy * \cylxx - \cylzx * \cylxy)}     \pgfmathsetmacro{\ang}{atan(\cylt)}     \pgfmathsetmacro{\ct}{1/sqrt(1 + (\cylt)^2)}     \pgfmathsetmacro{\st}{\cylt * \ct}     \filldraw[fill=white] (#4*\ct+#1,#4*\st+#2,#3) -- ++(0,0,#5) arc[start angle=\ang,delta angle=-180,radius=#4] -- ++(0,0,-#5) arc[start angle=\ang+180,delta angle=180,radius=#4];     \filldraw[fill=white] (#1,#2,#3+#5) circle[radius=#4]; }       \begin{tikzpicture}[x={(\xx*1cm,\xy*1cm)},y={(\yx*1cm,\yy*1cm)},z={(\zx*1cm,\zy*1cm)}]          \tdcyl{-3}{-3}{0}{1}{3} % x y z   r h         \tdcyl{0}{0}{0}{1}{3} % x y z   r h           \draw (-3,0,0) -- (3,0,0) node[circle,fill=white] {x};         \draw (0,-3,0) -- (0,3,0) node[circle] {y};         \draw (0,0,-3) -- (0,0,3) node[circle] {z};     \end{tikzpicture}     \end{document}   I added another cylinder and tweaked the positions to fit what I need. This produces the following:      Which is almost what I want, except that I want to see the projection of that cylinder on the x--y plane, but unfortunately, the top cylinder hides the full circle. When I disable the top cylinder:      \begin{tikzpicture}[x={(\xx*1cm,\xy*1cm)},y={(\yx*1cm,\yy*1cm)},z={(\zx*1cm,\zy*1cm)}]          \tdcyl{-3}{-3}{0}{1}{3} % x y z   r h            \draw (-3,0,0) -- (3,0,0) node[circle,fill=white] {x};         \draw (0,-3,0) -- (0,3,0) node[circle] {y};         \draw (0,0,-3) -- (0,0,3) node[circle] {z};     \end{tikzpicture}   Now the projection is shown again:   But the top cylinder is missing...  How can I make the top cylinder transparent, or, alternatively, and even better; how do I create one long cylinder and emphasize its projection on the x--y plane?   BTW, what determines the position of the labels x,y,z in the above code? I want to reposition them so they won't overlap with the axis. Thanks in advanced! How to install the latest Arduino IDE? I have a song that Windows can't play because there is a question mark in the name of the file.  "Where Have All the Cowboys Gone?.ogg" // as an example   So I try to rename it and Windows complains whether I try it in Explorer or from command prompt.  Error I get when trying to copy, rename, or move is:     The Filename, directory name, or volume label syntax is incorrect   Is there a Windows way to force a rename in this case?   Update  I'll keep an eye on this question, but after 13 answers and many attempts (aside form 3rd party solutions) it seems that Windows can't do this (or at least my windows can't, no short names). So I'm accepting the answer which was my original solution anyway of using Linux. It would be nice to see Windows handle this somehow, so don't stop just because I've accepted this answer, the question still stands!