I wonder why in $\mathbf{R}^2$ the only nontrivial subspaces are lines passing through the origin? Why not some other function (parabola, etc.) passing through the origin? I understand that for $P$ to be a subspace, it has to contain null vector. I do not understand, why it has to be a straight line.
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?
Convert XML/HTML Entities into Unicode String in Python I'm doing some web scraping and sites frequently use HTML entities to represent non ascii characters. Does Python have a utility that takes a string with HTML entities and returns a unicode type?  For example:  I get back:  &amp;#x01ce;   which represents an "ǎ" with a tone mark.  In binary, this is represented as the 16 bit 01ce.  I want to convert the html entity into the value  u'\u01ce' Terms for collections of animals As I watched the murder of crows sitting on the line above my house this evening, I got wondering where all of the collective nouns for animals (pod of whales, gaggle of geese, pride of lions) came from and why we need so many.    If sheep can be a flock, why can't whales, geese, lions, and crows? How can I concatenate two arrays in Java? Undergraduate prerequisites for a Ph.d. in combinatorics I am a double major in mathematics and electrical engineering. Towards the end of my mathematics major, I began to develop an interest in combinatorics (specifically graph theory), and am seriously considering a Ph.D. in the subject.   Having developed the interest late, my senior thesis will be in a different topic (complex ODEs), but I would have completed one course in combinatorics/graph theory (an upper-level undergraduate course using a grad-level text - I expect a strong reference from the professor teaching this course).   The courses I have completed are basically both the undergraduate and graduate-level algebra sequence,  analysis in R and R^n (without a formal introduction to Lebesgue measure), discrete math (for EECS), applied probability, one course in numerical methods, PDEs, applied complex variables, and a "mathematical methods for sciences" course (focusing on special functions, variational calculus, and integral transforms).   I won't have the standard math major courses(i.e,  topology, measure theory, and number theory.) I plan to focus more on applications than theory in my combinatorics Ph.D.   Would additional coursework be necessary for a Ph.D. in combinatorics (specifically graph theory)? (I could potentially delay my degree by a year and spend time as a non-degree student at another university, or complete a non-thesis track master's degree before applying for a masters/Ph.d.). Any advice would be appreciated. Prevent applications from opening browser Is there a free application out there that will intercept requests from applications to open a URL and prompt me as to whether or not I want the URL to be opened? (For instance, after Adobe Reader installs, it opens its website to get feedback about the install process.)  Ideally, it would be lightweight software for just this task, but if a larger firewall/security app provides this feature, that would be acceptable.  I'm using windows 7 32-bit.  As stating in a comment below, the idea is that you don't want the browser to open (assume it's not already running), so an extension (e.g. for Chrome, ffox) wouldn't be relevant. How do I prove equality of x and y? If $0\leq x,y\leq\frac{\pi}{2}$ and $\cos x +\cos y -\cos(x+y)=\frac{3}{2}$, then how can I prove that $x=y=\frac{\pi}{3}$? Your help is appreciated.I tried various formulas but nothing is working. Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$.   I want to prove that $$ \frac{N(t)}{t} \rightarrow \lambda, \quad t \rightarrow \infty \quad \text{ a.s.}. $$  To prove this I received a hint that I should use the martingale law of large numbers which states that:  Theorem: (Martingale LLN)  Let $S= (S_n = \sum_{k=1}^n X_k)_{n=1,2,\ldots}$ be a martingale with respect to the filtration $(\mathcal{F}_n)_{n=1,2,\ldots}$. Then $\sum_{k=1}^n X_k/k$ converges almost surely on the set $\left\{ \sum_{k=1}^\infty k^{-2} \mathbb{E}\left[ X_k^2 \mid \mathcal{F}_{k-1} \right] &lt; \infty \right\}$. Hence $S_n/n \rightarrow 0$ a.s. on the same set.  Now to use this theorem I think I need to use Doob-Meyer to decompose $N(t)$ as the sum of a martingale and an increasing process. Which would be the martingale $M(t) = N(t) - \lambda t$ and increasing process $A(t) = \lambda t$. But how is $M(t)$ the sum of a martingale? And I do not know how this exactly relates to the thing I have to prove. Thanks for any help. Stripping seasoning from cast iron After reading , I want to try to re-season a skillet with flaxseed oil; I'm fairly certain I did a terrible job when I seasoned the skillet several years ago. Is it possible to do this with steel wool or vinegar? (I don't have a self-cleaning oven, so that method is out.) If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$. If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.   I want to use induction to prove this:  It is trivial when $G=p(n=1)$, assume $G=p^n$, $N$ is a normal subgroup in $G$ with order $p$. Since $G$ is p-subgroup, $G$ has a normal subgroup $H$ with order $p^{n-1}$,according to assumption, $N \subset C(H)$. But I don't know how to go on, is $C(H)=C(G)$?Can induction work in this case? Would I be scrutinized at the border for traveling to the US twice in a quick succession? I am a British citizen visiting the USA in early August for a two week vacation with friends. After the holiday I'm returning to my home in the UK for three weeks. After that I will return to the USA and will be working for my British employer under the Visa Waiver Program for a period of 80-85 days.  I have read already that there is technically no problems with this under ESTA rules, but should I limit this working period to a shorter time frame (e.g. 70 or so days) so that it does not negatively affect the decision made by the POE officer? Ideally I would love to be over in the states for the near-full 85 day period.  Any advice or previous experiences anyone can offer?  Some clarifications: As confirmed by the immigration attorney, as I am employed by my UK employer, it will be regarded as a business trip, the important part being that I will be working officially as a consultant for my UK employer just overseas for the allotted 85 or so days I'm having trouble finding something to clarify this. It's been mentioned in the Difficulty Class section of the DMG (238) that there should be a 50/50 chance that an average person can succeed at a DC 10 challenge, implying you have to beat it. However, I'd always thought that there's a 50/50 chance for an average person to hit another average person, but the chance is really 55/45, since you only have to equal their AC. How does item level and rarity affect the possible attributes? What are the use cases for different scoping constructs? Deleting array elements in JavaScript - delete vs splice What is the best way to get the names of all of the tables in a specific database on SQL Server? Every retraction is a quotient map? I have to proof that every retraction is a quotient map.. I have no idea where to start or what to use! A retraction $r:X \rightarrow A$  is a continuous map s.t. $r(a)=a$ for every $a\in A$. Will I still receive OTA updates after rooting my Nexus 5? Is there any way to find out what happened to a post card sent from other country? Launch application on a specific monitor I have Ubuntu 14.04 and two monitors. When I open applications I need some of them to be on the primary monitor and others On the secondary. At the moment, apps always open in the primary monitor.   I have dash icons only on one (left) display, so I can't launch apps using the cursor or active window setting from Compiz. How to find all the available tools in mac terminal? I am trying to setup a bunch of development projects in my Mac. I have to run various tools like python, ruby, scala, groovy etc.  To see if a tool is available I use options like   ruby --version.    Instead I would like to see all the list of tools available in Mac terminal via a single command. What is the command/options for it?