Constraints :  Basis and dimensions are not introduced yet Matrix and system of equations can't be used  I am aware that this might possibly be a duplicate of the , but I wanted to take this question further and see if any of you have come across an alternative proof than the one mentioned in the first answer. Btw, I am new to this platform, so in case I am making a mistake by posting this, please let me know.
So I am in an introductory linear algebra(with proofs) class, and I came across the following theorem:  If $S$ is a subspace of a vector space $V(F)$ and is generated by $\{a_1,...,a_n\}$ then any subset $\{b_1,...,b_m\}$ of $S$ with $m&gt;n$ is linearly dependent.   The proof in the textbook is done through induction (as most proofs regarding finite sets are) but I really do not follow it and I feel there is a simpler way of doing it. A hint would be appreciated.
How is $\ln(-1) = i\pi$? How to change ubuntu install from legacy to uefi Shoenfield introductory mathematical logic book alternatives I was originally intending on reading Mathematical Logic by Joseph R. Shoenfield (1967) though after reading  I get the impression that despite being influential for the time it was written, there now exist many new books written from our current perspective that tend to be easier on non-experts by offering more thorough explanations for each new topic being introduced.   As of now I have a basic knowledge of abstract algebra (basics on vector spaces, rings, groups etc.) from doing elementary number theory and taking a course on linear algebra, also I have completed  a year long course on real analysis which used a lot of elementary set theory and some order theory.   With all that said, I was hoping based on my current basic knowledge of math that someone could recommend me an introductory book on mathematical logic which I could use as an alternative to Shoenfield's and which addresses Peter Smith's criticisms - any help would be greatly appreciated. How should I deal with becoming discouraged as a graduate student? How do I add a .sty file to my MacTeX/TeXShop installation? How to forbid keyboard shortcut stealing by websites in Firefox Many websites, especially everything involving rich text editing (this site is guilty as well), steal keyboard shortcuts normally used to control Firefox and make them do something else instead. It is totally infuriating when I press something like Cmd-number, Cmd-L, Cmd-T, or Cmd-K and it doesn't do what I want it to. Can I make it stop?  Actually, it would probably be for the best if I could forbid stealing of all Cmd-* shortcuts. I've never seen them used for anything useful. Is it possible? Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. Is the following true? Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.  I guess that it is true but I am not sure how to prove this. How do I find the current machine's full hostname in C (hostname and domain information)? In a C project (POSIX), how do I get the fully qualified name for the current system?  For example, I can get just the hostname of my machine by doing gethostname() from unistd.h. This might give me machine3 in return, but I'm actually looking for machine3.somedomain.com for example.  How do I go about getting this information? I do not want to use a call to system() to do this, if possible. Error -504 on appliation installation Yes. I've read all other questions regarding this problem. Like this one:  It's the same problem as this one but with more background information and an other app:   The details: My device is an Ulefone Be Touch 2 running Android 5.0.  My connection is perfect. To be sure I tried different networks, among them are 3G, 4G and wifi on ADSL, VDSL, cable and fiber. I've also used Tor and 3 different VPN providers in 3 countries on 2 continents. Let's say I ruled the network out.   A single app does not install: OpenKeychain.  Currently I have 126 apps installed. It does not install via desktop browser > play store > install button.    I have enough free space (40GB). The app is 7MB.   Re-adding Google account and clearing caches did not work.    Even a complete device wipe and re-install didn't work.     Downloaded the apk and tried manual installation, same error.   I also own a Nexus 7 (2013 wifi) and the app installs fine on that device. An idiom meaning someone's doing something useless and has no result at the end In my native language, we use an idiom to warn someone that they're doing something which has no result at the end:     Trying to convince him is like squashing water ...   Is there any idiom in English with the same meaning? The equivalence of sample correlation and R statistic for simple linear regression Is the solution to a question regarding four smaller masses on a rotating hollow sphere accurate? I was doing the question :     A uniform 8.40-kg spherical shell, 50.0 cm in diameter has four small   2.00-kg masses attached to its outer surface and equally spaced around it. This combination is spinning about an axis running through the   center of the sphere and two of the small masses (see ). What friction torque is needed to reduce its angular speed   from 75.0 rpm to 50.0 rpm in 30.0 s?   I understand how to get $\alpha$, and how to get $\tau$ from $I$ and $\alpha$. What I do not understand is their computation of the moment of inertia. The question did not provide any information regarding the shapes of the masses, but the solution (which writes $I=\frac{2}{3}MR^2+2mR^2$) seems to assume that the masses are point particles. Thus, the two masses along the rotational axis have a moment of inertia of 0 ($r=0$), and each of the other two masses has a moment of inertia of $I=mr^2 =mR^2$, where $R$ is the radius of the sphere and $r$ is the distance from the axis.  But why are the masses considered to be point particles? I wanted to use the parallel-axis theorem to determine the moment of inertia about the rotational axis; not knowing the shape of the masses made this impossible, since I couldn’t find the moment of inertia about the CM. If I knew $I_{CM}$, then couldn’t I use $I_{axis}=I_{CM}+mR^2$ for the two masses not on the rotational axis and $I_{CM}$ for the two that are? That would make $I_{total}=\frac{2}{3}MR^2+4I_{CM,\:mass}+2mR^2$ Would that not be a better way of solving this problem? Why doesn't my flash fire from the hot shoe trigger? I have a Yongnuo YN-560 III and a pair of RF-603II (C3) triggers. I have a Canon Rebel T5i.   I put one trigger on the hot shoe.  When this trigger is in TX mode I get a blue comm light and I can fire a test flash using the large test button on the trigger. But when I press the shutter button to take a picture, there's no flash.  The trigger and flash are on Channel 1. The trigger on the hot shoe is set to TRX. TX is fine, TRX gives me no flash. The flash and camera are in M mode.  I don't have a trigger on the flash because the Mark III has a built-in trigger The problem seems to be the camera.  The trigger can talk to the flash but the camera is not sending a flash command through the trigger in the hot shoe to the flash.  Grateful for anything you could suggest. How can I avoid "please insert CD/DVD" error when updating Ubuntu? I can't update 12.04 LTS. The Update Manager shows this:  CD/DVD 'Ubuntu 12.04 LTS _Precise Pangolin_ - Release i386 (20120423)' is required Please insert the above CD/DVD into the drive '/media/cdrom/' to install software packages from it.   The image I downloaded and burned to CD is from  the PC (Intel x86) desktop CD version. I burned many CDs but the result is the same. Two layers, same coordinate system, wont show up I have vector two sets of vector layers that won't show up together. One set shows up together and the other set together etc but they are of the same area.  The layers I was given did not have a defined spatial reference which I manually projected as the same as the data frame I specified so all layers now have same CRS...yet not showing up together. I have noticed they appear to have different extents... What should I do? Export all symbols when creating a DLL With VS2005, I want to create a DLL and automatically export all symbols without adding __declspec(dllexport) everywhere and without hand-creating .def files. Is threre a way to do this? Where in the world is tikzlibraryhobby.code.tex? I was trying to reproduce Claudio Fiandrino's answer to .  However, I do not have the hobby package by Andrew Stacey in my local distribution.  After reading  and , I looked at the latest  (build on December 7, 2012) but could not located the hobby package in it.  I found  but do not know where each file is supposed to go in my distribution.  How do I add the hobby package? In general, what would be the easy way to get any new TikZ package installed in a Mac?  EDIT:  I have used the drag-and-drop (mac) method as described below.  (1) Drag pml3array.sty file to /Users/Username/Library/texmf/tex/latex folder. (2) Drag hobby.code.tex file to /Users/Username/Library/texmf/tex/generic/pgf/math folder. (3) Drag pgflibraryfpu.code.tex to /Users/Username/Library/texmf/tex/generic/pgf/libraries folder. (4) Drag tikzlibraryhobby.code.tex to /Users/Username/Library/texmf/tex/generic/pgf/frontendlayer/tikz/libraries folder.  I no longer get the hobby cannot be found error message.  But, is this correct? I know that any group satisfying $x^2=1$ for all $x$ is abelian. Is the same true if $x^3=1$? I don't think it is, but I can't find a basic counterexample. How do I get "restart" option in the shutdown menu in Unity? The dropdown menu I get when I click on the gear link in the top right corner only shows these options:   Lock Computer Logout Suspend Hibernate Shutdown   There is no "restart" or "reboot".  How can I make that option available?  Bonus question: Can I remove Hibernate, Lock, and Suspend? Is the High Elf's extra cantrip in addition to cantrips from your class?