I've a problem with this: Let $f:V \to W$ be a linear function between two vector spaces, let $K$ be a subspace of $W,$ then $\dim (f^{-1}(K \cap \operatorname{im} f))= \dim (\ker f)+ \dim(K \cap \operatorname{im} f)$  How can I prove that? I can't understand this completely, it is similar to the rank-nullity theorem, is there any link between them? Thanks a lot in advice
Let $V,W$ be two vector spaces over a field $F$ and let $f:V \to W$ be a linear mapping. I have going over some facts about vector spaces (not necessarily finite dimensional) and I have been trying to apply the rank nullity theorem to derive a simple lemma.  I have not been able to complete the proof and was wondering if I need to use something other then $\ker f + \mathrm{Im}\; f = \dim V$.  How do we show that for ever vector subspace $K \subset W$, $ \dim(f^{-1}(K)) = \dim(K \cap \mathrm{Im}\; f) + \dim( \mathrm{Ker}\; f)$?
I've been trying a simple recipe I found on the internet, which is a simple recipe consisting mostly of gelatin.  The problem is that the end result is basically just a fruit-flavoured jelly (Jell-O?), rather than the more chewy gummy-bears (Haribo) style of jelly confectionery.  Can anyone suggest what I need to do to make them less springy and more chewy.  It's all difficult to articulate!  Here's what I used for the first test-batch:   12 tbsp sugar 12/3 cup fruit juice 8 tbsp Golden Syrup 8 tbsp gelatin   soften the gelatin in 1/2 cup of cold water. place sugar, syrup and the juice in a pan and gently heat until sugar dissolves. Stir in the gelatin and stir until dissolved. Pour into moulds and wait until set.    Thanks Completeness and Incompleteness Through browsing questions asked in the past about Godel's completeness and imcompleteness theorems, I've come to see that the sense of completeness in both of the theorems are different. However, I can't see how they are different!   Does this distinction boil down to that of truth and proof? Is it the case that the completeness theorem talks about the relationship between truth and proof, while the incompleteness theorems only relate to provability?  Could someone please elaborate on the sense in which both notions of completeness are different? $m$ is a perfect square iff $m$ has an odd number of divisors? This weekend is a "Supermoon" (the largest full moon of the year, in perigee, the closest to Earth). Would someone please advise how I could make the most out of my Canon PowerShot to photograph this phenomenon?    What settings should I use?    What times are the best with respect to the exact times of moonrise and moonset provided by the U.S. Naval Observatory? In other words, is it better 30 min before it crosses the horizon? Or just at the crossing? Are there any tips and tricks for a beginner to benefit from?  Are moonrises or moonsets generally better/easier to capture? Are there different techniques for different phases of the moon?   I read that full moon tends to come out flat, but I also want to benefit from the perigee. Can "whatever" be split into two words? I tend to write, "say whatever they want", but I'm always tempted to write "say what ever they want". Is it acceptable to split the word in this context? I have read that a holomorphic function in an open disc has a primitive in that disc. Does this mean that for any holomorphic function $f$ in any open set $U,$ that $f$ has a primitive? I say this because I can take a large enough open disc to cover the open set $U$ and find its primitive there. Is this a correct way of thinking? On OpenBSD and with OpenBSD's  and , is it possible to store the  of a user in a  repository of the same user?  What would be the proper way to accomplish something like this?  (To steer the answer into the correct direction, I would not be opposed to changing some permissions in the system, although I'd rather not have to recompile the binaries, nor violate any good security paradigms.) Spam iCloud Calendar Invitations Today I've suddenly received spam calendar invitation for $19.99 Ray Ban sunglasses (never subscribed, checked, searched or visited any related site, so clearly spam) and it has other visible email accounts within the invitation. I'm wondering if I should be worried whether my accounts been compromised or a simple email from anywhere just plops this in my calendar without review. Questions to be answered:  What should I check/setup in order to make sure my iCloud account is safe? How can I delete this invitation without sending a notification to the sender, so as to prevent them from knowing my email address is live? How can I prevent these notifications in the future?  Here's an image: Lebesgue integral $\int_{(0, \infty)} \frac{\sin x}{x} dm$ doesn't exist but improper Riemann integral exists My laptop's cable broke and sadly it seems like I need to buy a new power supply.  What I wonder is, I've found that universal power supplies are much cheaper than the official HP ones. Are they really worth it?  Another question: the original supply has these specs: 65W, 18.5V, 3.5A. However the universal supplies are more powerful (90W, 100W) and come in integer voltages (17V, 18V, 19V, etc). Are they compatible? Should I set them to 18V or 19V? Is there a function to extract the extension from a filename? LAN not working on 17.04 Actually yesterday I've upgraded from Ubuntu 16.04 to 17.04 and I got a problem that, if connect LAN cable it does not showing anything and  remains disconnected. But i can connect with WiFi. I don't know what's the problem. Actually its showing "device not managed" under the "Ethernet Network" and both were disabled. I've searched for the solution, but did not found yet. I've seen videos of some folks having several armor sets (4 is the most I've seen so far) standing about on their settlement. I'm wondering if having too many would cause some sets to disappear. They're not like guns/items that you can safely store in settlement crates, the suits kind of just... stand around. (I know there's the extra step of taking out the fusion cores, so that NPC's don't use them.)  Is there a limit to the number of Power Armor sets you can keep? How can I respawn the Ender Dragon? I have a server and the Ender dragon has been defeated.  However, returning to The End does not respawn the Ender Dragon.  How can I get the final boss back on a vanilla server without using mods? Modifying Score text in Locate Widget of Flexviewer 2.4? I want to add text where it says "score" to make it more clear that this is an address matching score.    I am using the LocateWidget_US.xml and do not think I can do it here.  I think the place to do it may be in the LocateWidget.mxml in this line????   locateResult.content = getDefaultString("resultScoreText") + " " +   (addrCandidate.score % 1 == 0 ? addrCandidate.score : addrCandidate.score.toFixed(1)); Prove that field of complex numbers cannot be equipped with an order relation Please guide me in this problem. I am confused about whether its asking that having the relation $z&gt;0$ does not satisfy the order axioms.  Any help would be really appreciated.  Thanks! Having links relative to root? Website is under maintenance - how to restrict access? Cauchy-Schwarz inequality proof (but not the usual one) Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the inequality (for a proof of that see:  )    I am reading a book that claims the Cauchy-Schwarz inequality is actually: $$\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert$$ where $\Vert x\Vert :=\sqrt{\langle x,x\rangle}$  with the additional claim: equality holds $\iff\ x,y$ are linearly dependent  I cannot find a proof of this claim (only proofs for the dot product inner product). My question is: what is the simplest way to prove this (I define simplest to mean "using as few definitions outside of the inner product itself as possible" - so that's not including Gramm-Schmitt orthonormalisation, for example.)  I've just found some possible answers in the links on the right (annoyingly) but fortunately I have a second, albeit softer, question:  What inequalities that are common for say $\mathbb{R}^n$ are actually based on the inner product?   (In future I will check this site first, I was targeting lecture notes and such in my search)  Addendum: I want to use this on not-finite dimensional vector spaces. I've found one proof that relies on finite-ness. Is there an issue tracker for already reported Mathematica bugs?