Claim: The subspaces of $R^2$ are precisely {$0$}, $R^2$, and all lines in $R^2$ through the origin. My attempt: Proof. Suppose U is a subspace of $R^2$. We want to show that the only possible dimensions of $U$ are $0$,$1$, and $2$. Clearly $U$$=${$0$} whenever $dimU=0$. If $dimU=1$, then the length of any basis in $U$ is $1$. Thus any nonzero list of length $1$ in $U$ is a basis of $U$. Hence if $(x,y)$$\in$$U$ with $(x,y)$$\neq$ $(0,0)$, then every $u$$\in$$U$ has only one representation as a linear combination of $(x,y)$. In other words, there exists a unique $\lambda$$\in$$R$ s.t. $u=$$\lambda$$(x,y)$ $\forall$$u$$\in$$U$. Therefore, for any nonzero $(x,y)$$\in$$U$, we have $U$={$\lambda$$(x,y)$ : $\lambda$$\in$$R$}, which is the set of all lines in $R^2$ through the origin. To show that $U=V$ whenever $dimU=dimV$ and $U$ is a subspace of $V$, let $u_1,...,u_n$ be a basis of $U$. Because $V$ is finite-dimensional and $U$ is a subspace of $V$, we can think of $u_1,...,u_n$ as a linearly independent list in $V$. Thus $U\subset$$V$. Now since $dimU=dimV$, it follows that $u_1,...,u_n$ is a linearly independent list in $V$ with length $dimV$. This implies that $u_1,...,u_n$ is a basis of V. Thus $V\subset$$U$ and hence $U=V$. The result above shows that if $dimU=2$ (i.e. $dimU=dimR^2$), then $U=R^2$. Because $dimU$$\le$$dimR^2$, it follows that these are the only possible dimensions of $U$. Therefore, the subspaces of $R^2$ are precisely {$0$}, $R^2$, and all lines in $R^2$ through the origin, as claimed.
I'm having trouble with a question from an introductory Linear Algebra book. It goes: "Show that the Only Subspaces of $R^2$ are the zero subspace, $R^2$ itself, and the lines through the origin." I'm thinking the easiest way to do this is to show that if $W$ is a subspace of $R^2$ containing $2$ different lines through the origin then $W$ is all of $R^2$. Is this a good way to go about it? and how could I show this?
Making separate Python installation that can call ArcPy? Sharing alignment between equations in two different items In the following minimal example, I have two equations inside an itemize environment, with an item per equations. Each equation is defined in its own equation environment.    \documentclass{article} \usepackage{amstext}  \begin{document}  \begin{itemize}  \item Lorem ipsum dolor sit amet, consectetur adipiscing elit:   \begin{equation}     f(x) = \left\{     \begin{array}{l l}       1 &amp; \text{ nisl justo, hendrerit} \\       0 &amp; \text{ sagittis condimentum}\\     \end{array} \right.  \end{equation}   \item Duis magna nunc, ultrices at fringilla non, feugiat sed massa:   \begin{equation}     g(x) = \left\{     \begin{array}{l l}       -1 &amp; \text{ fermentum fringilla mauris eget} \\       0 &amp; \text{ gravida ipsum vitae}\\     \end{array} \right.  \end{equation} \end{itemize}  \end{document}   However, each equation has its own alignment, giving the following result:    It would be nice (I think) if both equations could be aligned in the same way, something like this:    Is this possible?   Note: it would be nice if the solution could be also applied to equations in different paragraphs.    UPDATE:   presented a very nice solution, which is currently the most upvoted, but it does not work quite well:  \documentclass{article} \usepackage{amsmath} \begin{document}  This is a list.  \begin{itemize}    \item First item.   \begin{align*}   f(x) &amp;= \begin{cases} 0 &amp; \text{now} \\ 1 &amp; \text{then} \end{cases}   \intertext{\item Second item.}   g(x) &amp;= \begin{cases} 0 &amp; \text{here and there}  \\ -1 &amp; \text{to and fro} \end{cases}   \end{align*}  \end{itemize}  After the list. \end{document}     Can you see? "now" and "then" are not aligned to "here and there" and "to and fro". Can we solve this? :) What does “times” mean in the newspaper “Washington Times”?  I have always wanted to ask about this, but feared it might be so obvious that I would look like a fool. For that many times, I haven't before experienced that level of hesitation with any other question, but I think now is the right time to ask, anyway. Do paths in Adobe Photoshop have to be closed? I am working to convert a png to a vector graphic.  I am outlining the image using the pen tool.  Photoshop insists that the pen tool produce closed loops in order for me to use the "Stroke path..." tool.  Why is this? Apparently ;-) the standard containers provide some form of guarantees.  What type of guarantees and what exactly are the differences between the different types of container?  Working from  (about ) I have come up with this:  Container Types: ================ Container:     Forward Container         Reverse Container             Random Access Container     Sequence         Front Insert Sequence         Back  Insert Sequence     Associative Container         Simple   Associative Container         Pair     Associative Container         Sorted   Associative Container         Multiple Associative Container  Container Types mapped to Standard Containers =============================================  std::vector:    Sequence    Back        Sequence                    Forward/Reverse/Random Container std::deque:     Sequence    Front/Back  Sequence                    Forward/Reverse/Random Container std::list:      Sequence    Front/Back  Sequence                    Forward/Reverse Container std::set:       Sorted/Simple/Unique    Associative Container       Forward Container std::map:       Sorted/Pair/Unique      Associative Container       Forward Container std::multiset:  Sorted/Simple/Multiple  Associative Container       Forward Container std::multimap:  Sorted/Pair/Multiple    Associative Container       Forward Container   Container Guarantees: =====================                                                                                    Simp                                                                                   or                           For   Rev  Rand        Front  Back  Assoc        Sort   Mult                     Cont: Cont: Cont Cont: Sequ: Sequ:  Sequ: Cont:        Cont:  Cont: Copy    Const:      O(n) Fill    Const:                             O(n) begin()             O(1) end()               O(1) rbegin()                        O(1) rend()                          O(1) front()                                    O(1) push_front()                                     O(1) pop_front()                                      O(1) push_back()                                             O(1) pop_back()                                              O(1) Insert()                                                                          O(ln(n)) Insert: fill                               O(n) Insert: range                              O(n)                                   O(kln(n)+n) size()              O(n) swap()              O(1) erase key                                                     O(ln(n)) erase element                                                 O(1) erase range                                                   O(ln(n)+S) count()                                                       O(log(n)+k) find()                                                        O(ln(n)) equal range                                                   O(ln(n)) Lower Bound/Upper Bound                                                    O(ln(n)) Equality                  O(n) InEquality                O(n) Element Access                       O(1) Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$ "Give a hand" vs "Help" Currently, I'm working in an international company with many non-native speakers (so am I) but with pretty good level of English, I suppose.  I have noticed that my colleagues usually use phrase "give a hand" when asking for  help or even talking about it and seems to be avoiding the strong word "help".  It looks a bit confusing to me especially after I made people laugh when I said  "I will help you" in one of our conversations, don't remember exact context though...  So could you please describe the difference between them? And is there something wrong or hidden meaning with help-based phrase? Did the Big Bang happen at a point? Is there a way of setting culture for a whole application? All current threads and new threads? Is there a way of setting culture for a whole application? All current threads and new threads?  We have the name of the culture stored in a database, and when our application starts, we do   CultureInfo ci = new CultureInfo(theCultureString); Thread.CurrentThread.CurrentCulture = ci; Thread.CurrentThread.CurrentUICulture = ci;   But, of course, this gets "lost" when we want to do something in a new thread. Is there a way of setting that CurrentCulture and CurrentUICulture for the whole application? So that new threads also gets that culture? Or is it some event fired whenever a new thread is created that I can hook up to? I've been to South Korea a few times already and always enjoy it but I've always had an onward ticket before.  Now I need to fly home from a trip that's lasted over a year, but I still want to take my time and keep my options open.  There's a good choice of cheap flights from Istanbul to Seoul and from Seoul I can't decide yet whether to fly straight home or take a ferry to China, hitchhike to Guangzhou, and get a cheap flight home from there.  I can't really just buy the China-Australia flight now because I'm not sure yet whether I'll be able to get the Chinese visa easily enough or how long my remaining money will last.  I know Korea, like most countries, has rules saying you need an onward ticket, but in my experience many countries never try to verify that you have this ticket unless you fit some profile. So in this instance I would find first-hand knowlegde much more informative that pointers to a government website.  In practice does South Korea admit tourists with one-way tickets? Shortest code to calculate list min/max in .NET I'd like something like  int minIndex = list.FindMin(delegate (MyClass a, MyClass b) {returns a.CompareTo(b);});   Is there a builtin way to do this in .NET? "Closed" form for $\sum \frac{1}{n^n}$ Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.  One series that popped into our discussion was $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$.  Proving the convergence of this series is trivial but finding the value to which converges has defied me so far. Mathematica says this series converges to $\approx 1.29129$.  I tried Googling about this series and found very little information about this series (which is actually surprising since the series looks cool enough to arise in some context).  We were joking that it should have something to do with $\pi,e,\phi,\gamma$ or at the least it must be a transcendental number :-).  My questions are:   What does this series converge to? Does this series arise in any context and are there interesting trivia to be known about this series?   I am actually slightly puzzled that I have not been able to find much about this series on the Internet. (At least my Google search did not yield any interesting results). This questions has been asked before too and I did go through most of the suggestions. Things I have tried so far: setting up correct  path variable environment, installing anaconda 32 bits, python 2.7, or even switching to python 2.7 in cmd while initially using Anaconda 3 , adding Desktop 10.4.pth file with  following path.  C:\Program Files (x86)\ArcGIS\Desktop10.4\bin C:\Program Files (x86)\ArcGIS\Desktop10.4\ArcPy C:\Program Files (x86)\ArcGIS\Desktop10.4\ArcToolBox\Scripts   I also added those paths in environment variable   These suggestions help me to reduce the error. But Now I am getting a different error. Looks like python can find arcpy but missing another package.  Getting this error while trying to import arcpy in python 3.6.   File "C:\Program Files (x86)\ArcGIS\Desktop10.4\ArcPy\arcpy\__init__.py", line 22, in &lt;module&gt;     from arcpy.geoprocessing import gp    File "C:\Program Files (x86)\ArcGIS\Desktop10.4\ArcPy\arcpy\geoprocessing\__init__.py", line 14, in &lt;module&gt;     from _base import *  ModuleNotFoundError: No module named '_base'   Any suggestions? User  answered the wonderful  question by mentioning the using keyword. Can you elaborate on that? What are the uses of using? How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$? Chrome address bar no longer shows protocol or www subdomain A couple of days ago I noticed that for all websites, chrome omits from the address bar the protocol and the subdomain if it is www. My co-workers report that they are not seeing this. Chrome version is 67.0.3396.87.  For example, if I type into the address bar https://www.google.com (without the quotes), Chrome immediately displays the chip "Secure" followed by google.com. See attached screenshots.      I have looked in vain for Chrome settings that affect this, plus the chrome://flags page and chrome://net-internals page.  I did not do anything consciously to make this change.  Any ideas?  EDIT: Suggestion from MrWhite that Ominbox UI Hide Stead-State URL Scheme and Trivial Subdomains had become enabled: good suggestion, except that the setting is still at “Default.” See screenshot:    However, as a shot in the dark, I am going to play with the setting to see what happens.  First I set it to “Disabled.” That restored the previous behavior of the address bar, which is to not to hide the protocol and “trivial subdomains.” Then I changed it back to “Default,” and it was hiding them again. One would suspect that somehow the default has been changed, except that co-workers are not reporting this issue.  Oh, well. I’ll just leave it as Disabled for now.  I would be ready to accept the answer from MrWhite, because it provides a workaround. Yet will leave it open for a bit longer, because it does not really answer why this behavior has changed, and why it has to be overridden.  EDIT 2: Chrome version is 67.0.3396.87. interaction between mathematical structures From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures "interact" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one... How do I design a 2A or more power supply for my consumer USB devices? I understand that many of my consumer devices charged by USB ports can charge at higher rates than 0.5A.  However, in testing, I've found they don't consume more than this. Test Setup  I have a power supply that outputs between 0 and 30 volts, and 0 to 20 amps. I set the voltage to 5 volts, then connected up a female USB to the wires correctly. When I plug in a Samsung Galaxy S4 Mini, it pulls around 0.44 amps. It is the same with a Samsung Galaxy Tab Pro. However, when I connect an iPad Air with an official lightning cable, it only pulls around 0.11 amps. Also, I tried connecting a ZGPAX S28 smartwatch and it was still about 0.44-0.45 amps.  The power supply is more than capable of outputting the maximum 2.1 amps, so why doesn't it to at least the tablets? What do I need to do in my test setup to convince the devices to consume their maximum charging current? Maclaurin Expansion for $e^{e^{z}}$ at $z=0$ How to control internet access for each program? I would like to use a software to control which program may connect to the internet. I know that this behaviour is associated with the word "firewall", but some Linux users are very upset if somebody demands a Personal Firewall. I don't want to upset you by demand such a program. I don't want to "secure ports" or other stuff a Personal Firewall promises on Windows. I looked into iptables but it does not fit my requirements.  I saw an excellent answer here () but it's very uncomfortable to set this up.  Is there a software that asks for each program if it may access the internet?