This is an example from Linear Algebra Done Right by Axler:  Suppose that $U = \{(x, x, y, y) \in F^4 : x, y \in F\}$ and $W = \{(x, x, x, y) \in F^4 : x, y \in F\}$ Then $U + W = \{(x, x, y, z) \in F^4 : x, y, z \in F\}$  I don't understand how $U$ and $W$ sum to that. Can someone explain how this conclusion came about? Thank you!
Suppose that we have a vector space $V$ and the subspaces $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$. Then how comes $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}?$$
"On/at/for/over the weekend" in American English Some sources say that "at the weekend" is wrong, while other ones say it's correct. Which form is acceptable in American English?        On Saturdays her sister Ann usually comes to stay with Mary on/at/for/over the weekend/s.   What are you going to do at/on/for/over the weekend/s?    We are going to Paris on/at/for/over the weekend/s.   Are you going to stay here for/at/on/over the weekend/s?      Which preposition do American English speakers use — at, on, for, over? House address is not listed on Google Maps When I search for my address on Google Maps it cannot be found which means I can't order takeaway etc.  Is there a way of adding my address to Google Maps or even add a street address?  I want the address to be publicly available so others can search it. Is it possible to install Ubuntu on my Windows 8 computer without any CD or USB drive? I posted this question but for some reason I can't get back into my account:    Most things I see with 220V have both hot legs going through the switch or relay, current "returns" through the switch.  Is it against code to wire one leg directly to the equipment and the other leg is switched as opposed to having both legs switched.  It seems useless and a waste of wire to switch both legs.  Basically  vs  (minus the GFCI as I'm not allowed to have that)  As a bonus question.  I assume I need a "motor rated switch" as standard switches can weld due to the high startup current.  Is  one legal\safe? How to add indicator on a light switch to indicate the outdoor 3-way light is on? Have some carriage lights outside my garage that can be turned on from inside the house. It's common that the switch will get flipped and the lights will stay on when they're not wanted.  Is there some way I can add an indicator to the light switch so I can know if the light is on?  Note: the carriage lights are on a 3 way switch. Search for a string in all tables, rows and columns of a DB How do I get adequate depth of field in macro photography? I have been messing around with some Macro Photography using the techniques (and a reversal ring) described in a  by @ My main issue is that I am having a hard time getting the subject into focus. There are times when the center is blurred and the outside is in focus or the center in focus and the outside out of focus. What are ways to get as much of the subject in focus?   I am using a Nikon D5100, I have a 18-55mm kit lens but I find the 55-200mm lens easier to use for shooting. Showing group with $p^2$ elements is Abelian I have a group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, and with that I was able to show $G$ has a normal subgroup $N$ with $p$ elements.   My problem is showing that $G$ is abelian, and I would be glad if someone could show me how.   I had two potential approaches in mind and I would prefer if one of these were used (especially the second one).   First: The center $Z(G)$ is a normal subgroup of $G$ so by Langrange's theorem, if $Z(G)$ has anything other than the identity, it's size is either $p$ or $p^2$. If $p^2$ then $Z(G)=G$ and we are done. If $Z(G)=p$ then the quotient group of $G$ factored out by $Z(G)$ has $p$ elements, so it is cylic and I can prove from there that this implies $G$ is abelian. So can we show theres something other than the identity in the center of $G$?  Second: I list out the elements of some other subgroup $H$ with $p$ elements such that the intersection of $H$ and $N$ is only the identity (if any more, due to prime order the intersected elements would generate the entire subgroups). Let $N$ be generated by $a$ and $H$ be generated by $b$. We can show $NK= G$, i.e every element in G can be wrriten like $a^k b^l $. So for this method, we just need to show $ab=ba$ (remember, these are not general elements in the set, but the generators of $N$ and $H$).   Do any of these methods seem viable? I understand one can give very strong theorems using Sylow theorems and related facts, but I am looking for an elementary solution (no Sylow theorems, facts about p-groups, centrailzers) but definitions of centres and normalizers is fine. How do I get started working with JSON in Apex? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, navigate to the  tag.  This question and answer is owned by the community! If you can improve it and make it more useful, you're welcome to edit it.  I have a specific JSON structure in mind. I need to either deserialize this structure that's coming in from a web service, or I need to serialize data into this structure to transmit it to another system. How should I get started on this project in Apex? What are the tradeoffs involved in taking a postdoc at a top university versus an assistant professorship at a non-top university? Is everything in the physical world composite? In philosophy there is a principle that anything composite cannot have existed eternally, since it is preceded by its parts and whatever forces assembled it.  Is everything in the physical world composite in one sense or another or is there something not composite? Pokemon keep running away and Pokestops don't drop anything when spinning. What is going on? I can't do anything now. Can't catch Pokemon, can't get items. Does anyone have the same problem? mailto link with HTML body Transpose of inverse vs inverse of transpose Real-time updates for new review tasks I've got a huge list of date-times like this as strings:  Jun 1 2005  1:33PM Aug 28 1999 12:00AM   I'm going to be shoving these back into proper datetime fields in a database so I need to magic them into real datetime objects.   This is going through Django's ORM so I can't use SQL to do the conversion on insert. Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements I am attempting to prove what the title says, that $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements.  I have already proven that for any field $K$ and polynomial $f(x)$ in $K[x]$, $K[x]/\langle f(x)\rangle$ is a field if and only if $f(x)$ is irreducible in $K[x]$. So I know that $\mathbb{F}_p/\langle f(x)\rangle$ is a field for sure. Where I'm lost is figuring out how to know for sure that there are $p^n$ elements. It's fairly clear if $n\in\{ 0 ,1\}$, but I don't think induction is going to help.  I know from a proposition in class that since $F_p = \mathbb{Z}/\langle p\rangle$ and $F_p = \mathbb{Z}/\langle p\rangle$ is an integral domain, $F_p[x] = (\mathbb{Z}/\langle p\rangle)[x] = (\mathbb{Z}[x])/(\langle p\rangle[x])$, but I'm not sure whether or not that's helpful information. How many sequences of $n$ tosses of a coin that do not contain two consecutive heads have tails as the first toss? If you toss a coin $n$ times, there are $2^n$ possible sequences of heads and tails. Let $E_n$ be the set of sequences which do NOT contain two consecutive heads and $e_n$, the number of sequences in $E_n$.  Thus $E_3$ $=$ $\{$ $TTT$, $TTH$, $THT$, $HTT$, $HTH$ $\}$ and $e_3$ $=$ $3$.  How many elements of $E_n$ have $T$ as the first toss?  Please can anyone help me out here? Does the artificer's Enhanced Weapon stack on top of hexblade's Improved Pact Blade? I play a hexblade warlock with an Improved Pact Blade longsword. I like the idea of the eldritch tinkerer, who builds his patron's gifts into his gadgets, which fits in a warlock/artificer multiclass. The artificer's Enhanced Weapon infusion would be very nice with the warlock if it stacks with the Improved Pact Blade bonuses, but I can't tell if it does from the class descriptions. Do these features stack, or do I need to rethink this build? Story ID: Boy who is stuck at 5 years old with robot nanny Here's what I remember:  The boy is 5 years old, and has been for some time, because there's a bit in which he recalls turning 5 (celebrating his 5th birthday) many different times  There's a robot nanny in the apartment with him that he learned to disable so that he can hang out with the girl, who he isn't supposed to know. (In my mind the nanny bears somewhat of a resemblance to Muthr from the Wondla series... however I definitely read this story well before reading Wondla)  There's a neighbor girl who teaches him to read (I think she's like 9), and when they first meet she asks how old he is (5 is the last age at which growth/development can be successfully (and safely?) suspended) and he lies, saying that he's 6, but doesn't know why he does.  When a child whose aging process has been stopped learns how to read, they begin aging normally again, so the parents don't want them to learn to read.  The children who have stopped aging are kept apart from other children who grow normally-- they have separate playgrounds and the like  The injection/serum used to halt the aging process was developed by a scientist, who when he first shared his findings had a 5 year old son and injected him, then told the onlookers/potential buyers/other parents to come back in 20(?) years. When they did, their children were adults and no longer needed them or were ungrateful layabouts, and the scientist's son was still 5 years old, cute, needy, etc.  The main character has a discussion about how he wants to turn 6 with his parents and they agree but only because he's already learned to read and it's irreversible.  It's kinda dark and emphasizes the need that parents have to feel wanted. Please help! I remember it so vividly and I'd love to read it again!