I understand there is a property of invertible matrices that states: "a square matrix is invertible iff it can be written as a product of elementary matrices"   However i need help proving this and justifying it
I'm just starting to learn linear algebra and I understand Gaussion elimination but I don't understand this proof:    Is the heart of this the fact that all the while when I've been row-reducing matrices to reduced row-echelon form... I've actually been multiplying a matrix by a series of a elementary matrices?  What does the notation $[A\ \ 0]$ mean?  How is the equation $A = E_1^{-1}...E_k^{-1}$ arrived at?
Differential form is closed if the integral over a curve is rational number. The following problem comes from do Carmo's book Differential Forms and Applications, Chapter 2, Exercise 4:  Let $\omega$ be a differentiable 1-from defined on an open subset $U\subset \mathbb{R}^n$. Assume that for each closed differential curve $C$ in $C$, $\int_C \omega$ is a rational number. Prove that $\omega$ is closed.   Can anyone give some hints? My idea was that we could try to show the integral vanishes if the curve lies in $N_\epsilon (p)$ (i.e. a small neighborhood around $p$) for any $p$ in $U$ and use the Poincare Lemma, but failed. How to check type of files without extensions in python? Page Object Model or External property file for Selenium Object Repository I am a beginner in Selenium WebDriver (Java with TestNg). I'm trying to create a framework of my own.  My question here is what would be the better option to store my objects (basically all the webelements of my website under test), Page Object Model or External Property file? Why? Is there a better method than those? Please help. Register On-Click Event with jQuery - Not Firing Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ Find the limit: $$\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$$   I have no idea what to do, but I thought that this is the limit of the derivative of $f(x)=\left(1+x\right)^{1/x}$, as $x$ tends to 0. Any help? If $a \cdot a = 1$ then $a = 1$ or $a = -1$ BAV99 forward voltage I am reading the  From the table on ELECTRICAL CHARACTERISTICS (page 2) (TA = 25°C unless otherwise noted) (Each Diode)     Forward Voltage mVdc     IF = 1 mAdc 715mV   IF = 10 mAdc 855mV   IF = 50 mAdc 1.0V   IF = 150 mAdc 1.25V     On page 3 there is the following VI curve graph:  Clearly at 25°C - 100mA is Vf = 0.9V. There is huge difference between the table and the IV curve.  Which source of information I should trust? Is this (typical) and (maximum) where the table guarantees the diode won't exceed Vf 1.0v at 50mA while the graph is just typical values? What information does the trace of a matrix give? I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant will tell about the volume of the space enclosed by those vectors. Does trace give any information about the space? I have read about copying files with terminal but these examples will help me a lot. So here is what I want to do:  Examples:   I have a file in /home/levan/kdenlive untitelds.mpg and I want to copy this file to /media/sda3/SkyDrive and do not want to delete any thing in SkyDrive directory. I have a file in /media/sda3/SkyDrive untitelds.mpg and I want to copy this file to /home/levan/kdenlive  and do not want to delete any thing in kdenlive directory  I want to copy a folder from home directory to sda3 and do not want to delete any thing on sda3 directory and opposite I want to cut a folder/file and copy to other place without deleting files in that directory I cut it into. How to append the contents of a list at the end of the other list? How do I append the contents of one list at the end of another list? When to use \LetLtxMacro? Frequently I see users redefine macros using (for example)  \let\oldmacro\macro% Store \macro in \oldmacro \renewcommand{\macro}{%   % ...redefined \macro   \oldmacro% }   and other times they use  \usepackage{letltxmacro}% http://ctan.org/pkg/letltxmacro \LetLtxMacro{\oldmacro}{\macro}% Store \macro in \oldmacro \renewcommand{\macro}{%    % ...redefined \macro   \oldmacro% }   Why would one use \LetLtxMacro when \let works just as well? Or, more specifically, when would one use \LetLtxMacro and/or why? Can awakened mind be used with suggestion? Can the warlock class feature "awakened mind" be used in conjunction with the suggestion spell, allowing the warlock to make the suggestion telepathically?  Awakened Mind, Player's Handbook p. 110:     You can telepathically speak to any creature you can see within 30 feet of you. You don't need to share a language with the creature for it to understand your telepathic utterances.   Suggestion, Player's Handbook p. 279:     You suggest a course of activity (limited to a sentence or two) and magically influence a creature you can see within range that can hear and understand you.   If it works with suggestion, would it also work with mass suggestion?  To be clear, I am not asking about the verbal component of the spell, only the method the caster "suggests a course of activity". How to show google.com in an iframe? Serial protocol delimiting/synchronization techniques As asynchronous serial communication is widely spread among electronic devices even nowadays, I believe many of us have encountered such a question from time to time. Consider an electronic device D and a computer PC connected with serial line (RS-232 or similar) and required to exchange information continuously. I.e. PC is sending a command frame each X ms, and D is replying with status report/telemetry frame each Y ms (The report can be sent as response to requests or independently - doesn't really matter here). The communication frames can contain any arbitrary binary data. Assuming the communication frames are fixed-length packets. The problem: As the protocol is continuous, the receiving side might loose the synchronization or just &quot;join&quot; in the middle of an ongoing sent frame, so it just won't know where the start of frame (SOF) is. A the data has different meaning based on its position relatively to the SOF, the received data will become corrupted, potentially forever. The required solution Reliable delimiting/synchronization scheme to detect the SOF with short recovery time (i.e. it shouldn't take more than, say 1 frame to resynchronize). The existing techniques I am aware (and using some) of: 1) Header / checksum - SOF as predefined byte value. Checksum in the end of frame.  Pros: Simple. Cons: Not reliable. Unknown recovery time.  2) Byte stuffing:  Pros: Reliable, fast recovery, can be used with any hardware Cons: Not that suitable for fixed-size frame-based communication  3) 9th bit marking - prepend each byte with additional bit, while SOF marked with 1 and the data bytes are marked with 0:  Pros: Reliable, fast recovery Cons: Requires hardware support. Not directly supported by most of PC hardware and software.  4) 8th bit marking - kind of emulation of the above, while using the 8th bit instead of 9th, which is leaving only 7bits for each data word.  Pros: Reliable, fast recovery, can be used with any hardware. Cons: Requires an encoding/decoding scheme from/to the conventional 8-bit representation to/from 7-bit representation. Somewhat wasteful.  5) Timeout based - assume the SOF as the first byte coming after some defined idle time.  Pros: No data overhead, simple. Cons: Not that reliable. Won't work well with poor timing systems like, say, Windows PC. Potential throughput overhead.  Question: What are the other possible techniques/solutions exist to address the problem? Can you point to the cons in the above list which can be easily worked around, thus removing them? How do you (or would you) design your systems protocol? Is there any way to get a specific angle using the knife tool? Instead of the constrained 45 degrees that happens when you press C, is there a specific angle I can get for example 35 degrees instead?   It's currently cut at 45 degrees in the attached image and I need it to be 35. /A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand? How to extract pixel value counts from a raster in QGIS? How to grep for text in a file and display the paragraph that has the text? Below is the text in the file:  Pseudo name=Apple Code=42B state=fault  Pseudo name=Prance Code=43B state=good   I need to grep for "42B" and get the output from the above text like:   Pseudo name=Apple Code=42B state=fault   Does anyone have idea on how to achieve this using grep/awk/sed? Tense change: previous actions on something that's currently true I'm describing a situation that happened in the past. To explain it, I want to use a description that is both true now and true when the situation happened.  Specifically, I want something like:     She touched me where my neck met my collarbone.   Since my neck is still attached to my collarbone (thankfully), I'm wondering if I shouldn't use the present tense here instead:     She touched me where my neck meets my collarbone.   Which is preferable, and why? Can less retain colored output? Can I get less not to monochrome its output?  E.g., the output from git diff is colored, but git diff | less is not.