Given $A$ be $n×n$ matrix such that $A^{n}=0$, but $A^{n-1}$ not equal to zero a vector $v$ belongs to $\Bbb{R}^{n}$. Proof that {$V,AV,\cdots,A^{(n-1)}V$}  is a basis.
Let $A \in Mat_{n \times n}(F)$ a matrix which satisfies $A^n=0$ and $A^{n-1} \ne 0$ for some positive integer $n$. Let $v \in F^n$ be such that $A^{n-1}v \ne 0$.  Prove that $\{v, Av, A^2v,...,A^{n-1}v\}$ is a basis for $F^n$.     So, from definition of $A$, I know that $A^iv \ne 0$ for all $0\le i \le n-1$. I know I need to prove the set is linearly independent and span $F^n$ to prove it's a basis, but I just don't know how.
Does metric signature affect the stress energy tensor? I want to create an image like below, so a balloon with text on it.     I have been googling for ideas (text, sphere, ....) and came on  earlier question. So I tried Shrinkwrap and Cast modifier.   Here is the situation before applying the modifiers.    This was after trying Shrinkwrap    And this after Cast modifier.    For me it does not look like it is going to work with one of these modifiers.  Question How to create the image of a balloon with text on it ? With one of these modifiers or should I try a complete other approach ?? How to do this ? What effect does serial downvoting have on my rep cap? So I was the victim of serial downvoting the other day, and just had it automagically reversed. Rather than simply removing the previous votes, I was given a bonus lump of rep with the reason "serial downvote reversal". The downvotes and the bonus show up on different days. Since a whole day's quota worth of downvotes equates to quite a fraction of my daily rep cap, what happens in the following circumstances?   I get 200+ rep on a particular day, but I am serial downvoted (later reversed) back to under 200. Does this day count as reaching cap? I get 200+ rep the next day, and I am also credited back the rep. Is the rep bonus immune to the rep cap in the same way bounties are? On the day I get my rep bonus, it is enough to take me to 200 rep. Does this count as capping? How should I phrase a question that must be answered with an ordinal number (e.g., the third prime)? If I take a sponge and place it in a shallow dish of water (i.e. water level is lower than height of sponge), it absorbs water until the sponge is wet, including a portion of the sponge above the water level. In other words, it seems the sponge pulls some water from the bath up into itself, doing work, and the water gains some gravitational potential energy.  Where does the energy required to do this work come from? My suspicion is that the answer involves physical and/or chemical bonds between the water and the sponge, or possibly the change in the surface area to volume ratio of the water. How to download/read only the first 80KB of a file? How to extrude and scale with an even offset? I'm quite new to Blender, so forgive me if this is a dumb question. I don't even know hot to describe it properly, so:      Basically I want to extrude a rectangle and then scale it along two axis, but without keeping the proportions; I want to reduce them of the same quantity, so that I get something similar to the frame of a picture. Is it possible in Latex to make a triangular hat? The one that is used in math over three letters representing points to indicate a triangle.  Something like: build argument lists containing whitespace In bash one can escape arguments that contain whitespace.  foo "a string"   This also works for arguments to a command or function:  bar() {     foo "$@" }  bar "a string"   So far so good, but what if I want to manipulate the arguments before calling foo?  This does not work:  bar() {     for arg in "$@"     do         args="$args \"prefix $arg\""     done      # Everything looks good ...     echo $args      # ... but it isn't.     foo $args      # foo "$args" would just be silly }  bar a b c   So how do you build argument lists when the arguments contain whitespace? I tried to upgrade from Ubuntu 16.04 to Ubuntu 18.04. First, after downloading and installing of packages, I was unable to login. Login screen hanged completely and it showed Failed to authenticate. But then I logged into tty1 and then logged in Ubuntu. It was saying that I am in Ubuntu 18.04. Then it asked for reboot. So I rebooted Ubuntu from terminal. It got stuck at something like /dev/sda4 : Cleaning files. After trying for a few times, I got into tty1 and here I logged in as root user and tried to update and upgrade. But now I am getting errors like unmet dependencies and broken packages. I have done sudo dpkg -- configure -a. I don't want to do a reinstall of Ubuntu. How do I get access to my Ubuntu Desktop? Split a string by spaces -- preserving quoted substrings -- in Python I have a string which is like this:  this is "a test"   I'm trying to write something in Python to split it up by space while ignoring spaces within quotes. The result I'm looking for is:  ['this','is','a test']   PS. I know you are going to ask "what happens if there are quotes within the quotes, well, in my application, that will never happen. Which is correct? - "Guess, what it is?" or "Guess, what is it?" I want to know which sentence is correct. If i say     Guess, what it is?   or     Guess, what is it?   which one is correct? There was the following sentence in an article titled, “Like, Degrading the Language? No Way” in New York Times (April 5), in which the author says Americans are moving backward on language:     “(Like the use of ‘like’) the use of “totally” mines the same vein.   “He’s totally going to call you” does not mean “He is going to call   you in a total fashion.” It has a more specific meaning, although only   handled subconsciously by speakers, as so much of language is. “He’s   totally going to call you” contains an implication: that someone has   said otherwise or that the chances of it may seem slim at first glance   but in fact aren’t. As with “like,” “totally” tracks and nods to the   opinions of others. It’s totally civilized.”      Although the author explains “totally going to call you” contains an implication of someone having said otherwise, I’m not still clear with the meaning of “He’s totally going to call you.”   What does it mean? He’s totally going to call you “what”?   How does “He’s totally going to call you.” differ from “He’s going to call you.”? Is this expression (use of totally) ubiquitous today as the author asserts? If $|a| = 12, |b| = 22$ and $\langle a \rangle\cap \langle b\rangle \ne e$, prove that $a^6 = b^{11}$ Debugging with command-line parameters in Visual Studio I'm developing a C++ command-line application in Visual Studio and need to debug it with command-line arguments. At the moment I just run the generated EXE file with the arguments I need (like this program.exe -file.txt) , but this way I can't debug. Is there somewhere I can specify the arguments for debugging? User with low reputation shows as having modified thread How can this user , reputation 3, show in  as having "modified 7 hours ago", but on the modified thread, , not show at all?  This user also has two medals, which is somewhat puzzling. Why put a resistor in series with signal line? A lot of times in circuits I see a resistor placed in series in a signal line and sometimes even in series with an MCU's VDD line. Is the intention of this to smooth out noise in the line? How is this different from using a small cap, like a .1µF to do the same thing? I have read the thread regarding , but it does not answer my question.  I am confused about the notation for change in Physics. In Mathematics, $\delta$ and $\Delta$ essentially refer to the same thing, i.e., change. This means that $\Delta x = x_1 - x_2 = \delta x$. The difference between $\delta$ and $d$ is also clear and distinct in differential calculus. We know that $\frac{dy}{dx}$ is always an operator and not a fraction, whereas $\frac{\delta y}{\delta x}$ is an infinitesimal change.   In Physics, however, the distinction is not as clear. Can anyone offer a clearer picture? How can I backup my RAW photos while travelling without Internet access? This question seems similar to , but my needs are different :   I shoot in RAW exclusively I'm looking for a 100% offline solution: while traveling, Internet won't be accessible the "laptop" solution is too heavy for me: I'm looking for a lightweight solution the simplest, the better: one device to rule them all   EDIT:  I'm using good old Compact Flash cards (yes, the bigger ones), so I need a device that can read it. If it can read from other cards, this question might be interesting for more people than myself  - that's the whole purpose of the website.    Also asked by :  Portable Storage Device while traveling  I need storage device which is able to read CF cards and make a backup of my photos while traveling. I do not want to carry laptop (it's bulky) and tablet (too little storage) but looking into some other solutions.   So far I found:   Epson PXXXX - price is blocker for me Jobo Giga Vu - same as above Nexto DI - bit better, but still a lot Hyperdrive Colorspace UDMA - this looks the best for price/options Wolwerine PicPac - cheap, bad opinions DigitalFoci PhotoSafe - same as above Ex-Pro® Picture 2 Drive - didn't find much about it   So ladys &amp; gents what do you use? I read this story in the early 2000s, but it was a Chinese translation so the English version could be older.  The beginning is a bit like the Passengers movie: on a spaceship, a man woke up from stasis too early and found that he could not go back to sleep. So he has to spend his whole life on the spaceship, alone.  For some reason (technical or ethical), instead of waking other people up, he created a fantasy adventure story featuring a brave prince as a pastime. He did stories, and drawings all over the inside of the ship. Then he died, not knowing how people are going to respond to his work.