There is a problem in the book Linear Algebra by 'Hoffman Kunze':  Let $V$ be a vector space over the field $F$ and $T$ a linear operator on $V$. If ${T}^2$ = $0$, what can you say about the relation of the range of $T$ to the null space of $T$?   I was trying with $R(T^2)\subset R(T)$ and $N(T)\subset N(T^2)$ but couldn't get the answer....  any hint would be appreciated.....
Let $V$ be a vector space over a field $F$ and $T$ a linear operator on $V$. If $T^2$$=$ $0$, what can you say about the relation of the range of $T$ to the null space of $T$?
Converting truth tables to boolean expressions How can I automate the workflow for producing multiple versions of a document? Say you have two versions of the same LaTeX document.    For instance, you have one version of your lecture slides for actually delivering a lecture: as you're really not a big fan of the "wall of text", this version would be parsimonious in terms of text content, allowing the audience to listen to what you're talking  about instead of reading the slides, and giving graphical content centre stage.  Another version of those slides, made available to the audience after the lecture, for self-study, would be more self-contained and include a summary of what you said during that lecture along with the graphical content mentioned above.  Consider that you can toggle between the two versions by either changing a class option or by changing the value of some dedicated counter in your input file; something relatively simple to do "by hand".  To produce two different versions of the same document, my current workflow is   "manually" compile the first version, change the filename of the output produced (in order to prevent overwriting in step 4) make appropriate changes in my input file to switch to the second version, "manually" compile the second version.   However, this workflow is tedious and error-prone, especially when the number of different versions increases. I would like to automate it as much as possible. Is there a way to batch compile all the versions of my document, each getting a different output filename?  I guess that would involve modifying the input file at the command line (I'm on Mac OS X), but I don't know how to do that.  Do you have an automated workflow for compiling multiple versions of the same document? How do you do it?  Edit: I figured it out using  and  on , of which my question is a duplicate. Why are there so many close votes to review and so few of the others? Can the preposition 'in' be omitted here? As my question says,   $$\frac 1{1.2.3}+\frac 1{2.3.4}+\frac 1{3.4.5}+...$$  As far as $\frac 1{1.2}+\frac 1{2.3}+\frac 1{3.4}+...$ is concerned, we can write it as $\frac 1{n-1} - \frac 1{n}$ What can we do here? openSuse, new user cannot login I am using openSuse 12.3.  I have created a new user using:  linux-amvn:~ # useradd -m -G users,dev -s /bin/bash -p pass123 harbir-PC   The user harbir-PC is not able to login, when I try to log using ssh. I went ahead and looked into the /etc/passwd file and I see the following:  harbir:x:1000:100:harbir:/home/harbir:/bin/bash kdm:x:489:487:KDM Display Manager daemon:/var/lib/kdm:/bin/false harbir-PC:x:1001:100::/home/harbir-PC:/bin/bash   There is a difference between the user harbir and harbir-PC.  I have no problem with the user harbir (who I have created during installation), but I need to login using the user harbir-PC.  I also checked the /etc/shadow, and passwords are encrypted. {genealogytree} How to model my Cousins In MLE for continuous rv, why is it ok to evaluate a pdf at a point? Show/Hide function folding margin on Notepad++ Is it possible to get rid of the function folding margin on Notepad++? It seems like the sort of thing that should be in the prefs but I'm not seeing it anywhere. Should you use "who" or "that" when talking about multiple people doing something? Which of the following is correct?        There were 10 people that went to the store.   There were 10 people who went to the store.      Edit:  Which of the following is correct?        There were 10 people that had brown hair.   There were 10 people who had brown hair. What does a de Broglie wave look like? What does a  wave look like? Are de Broglie waves transverse or longitudinal?   Can they be polarized?    What about the de Broglie wave of a ground state neutral spin-zero Helium 4 atom?   What experimental evidence do we have that supports the detailed nature of a de Broglie wave?     I have always assumed that de Broglie waves were mathematically  identical to electromagnetic waves, but I just realized there is no basis for this assumption, and in fact it must be false, unless there is an analogue to both the magnetic and the electric components of the electromagnetic wave.  So what does a de Broglie wave "look like"? This is one of the few movies I saw on one of my relative's computer screen when I was little. As I was too young to understand anything, it was approximately 15 to 20 years ago (I believe the movie couldn't have come out after 2000, but the quality of the pictures looked good enough that I have my doubts).   The movie I am looking for was fully animated. Potentially a Japanese animated movie, but I can't tell for sure, as I was too young to tell the difference between Japanese, English or any language. There may be blood in the movie, but I can't tell for sure. However, now that I'm looking back, the drawing style looked slightly Japanese, although it was more of the bloody kind (I could slightly compare its style to "Blood: The Last Vampire" (2000)). Maybe it had an Asian tone to it (e.g. it could be a Korean animated movie too).   One of the "bad guys" was a tall, heavy guy, in a very armoured suit. I can't tell whether he was wearing a suit, or if his body was enhanced because he survived an accident (although I have some form of memory about the latter). Most of his body was covered, but I remember his face not being covered. The most distinctive feature he had was the fact that he had a Minigun where one of his arm should be (a bit like Barret in Final Fantasy 7). I think the first time we see him, he's seen crushing a metallic ashtray with his other hand.   A battle scene in the movie includes the heavy guy having a shoot-out against a relatively thin guy (or should I say "normal-sized man"), fighting him with a machine-gun mounted vehicle (he wasn't the one driving; he was shooting from the back of the vehicle). While they were shooting at each others, I remember the heavy guy screaming (nothing in particular, he was just shouting continuous "AAAHH~!!" in pure rage). The fight didn't seem to happen in a city. It slightly looked like a battlefield. The fight ended with some form of explosion, probably killing off the heavy armoured guy.   The other noticable character was a young boy. He looked... pale, almost weird. His skin was almost green, otherwise he looked like a regular child. He seemed to have some psychic powers too: at some point, when he was feeling uneasy and sweating, a man picked a sample of blood from the back of his neck, using a portable device. But when the man was... either incapacitated or down, that boy broke the device containing his blood simply by looking at it.   At some point near the end of the movie, the pale looking boy and some people (with scientists among them) seemed to be inside a huge facility or underground hangar. There was a lot of space, although I can't remember if it was a factory, or if it was an underground weapon storage. The child was also giving some form of speech too, as all the lights turned on suddenly. The others were possibly surprised by what they were seeing.   One last detail I remember about that child is his smile. While it wasn't as scary as the Joker's, his grin was... kind of creepy, because of how he looked at the others with his small round eyes. He was oftenly grinning.   Overall, most of the scenes I remember were quite dark (mostly because there was little light, or it was night time). The two exception being the battle scene (although it was dark, there was a lot of muzzle fire), and the last underground scene with the boy.   When I asked my relative about the movie, all they could tell me was: "The coloured word 'Aiko' was written on the cover of the CD". Why are more powerful tools using higher voltage? AFAIK each of the world leading cordless power tools manufacturers produces several "product lines" of cordless power tools with different voltages. For example, Bosch currently produces tools with Li-Ion batteries with 10,7V, 14,4V, 18V and 36V output and the higher the voltage the more powerful a tool is.  Now those tools are powered by batteries that are assembled from cells with lower voltage (something like 3,7 volts for Li-Ion cells I guess) and cells are connected in sequence until the target voltage is reached.  They could instead connect cells in parallel. They would have the same voltage, but higher current and that would again yield higher power.  Why do they choose higher voltage over same voltage and higher current to get higher power in electric tools? How to revert to factory settings? Alt Tags Missing from code Finding a 2x2 Matrix raised to the power of 1000 Duplicate questions on other SE sites? I recently asked a  on security.stackexchange.com. The question was related to XSS so in my opinion it belonged to the security website.  There was a comment on the question saying that a similar question exists on SO and my question was a duplicate (I think he gave it a down vote). The guy was right but the question was asked on another website (SO). There may be a case when person asking a question might not have an account with the other website.  Is it a good practice to mark questions duplicate if you have seen a similar question on some other website? JavaFX: Undecorated Window NSWindow Mac App Store like Title Bar The  currently says  In the context of Stack Exchange sites, any copying and pasting of any amount of text or code that wasn't written by you is plagiarism if you try, explicitly or implicitly, to pass it off as your own work. (italics mine)  There is also a concept of . Should this also be addressed in the FAQs? In other words, is it OK to copy (large parts of) your own answer  without proper attribution?