Let $V$ be a finite-dimensional inner product space, and let $E$ be a idempotent linear operator om $V$, i.e. $E^2 = E$. Prove que $E$ is self-adjoint if and only if $EE^*=E^*E$.
I have been trying to solve this problem for quite a while. I am still unsure of whether any of the avenues I have pursued have been of any use. Any advice will be much appreciated. Question: Let $V$ be a finite-dimensional inner product space, and let $E$ be an idempotent linear operator on $V$. Prove that if $EE^* = E^*E$, then $E$ is self-adjoint. (This is essentially exercise 5(a) in sec. 80 on p.162 of Paul R. Halmos, Finite-Dimensional Vector Spaces, but Halmos didn't assume that the dimension of $V$ is finite.)
The Problem For a small mass a distance $R_i$ away from the center of the Earth, how long would it take for the object to fall to the surface of the Earth, assuming that the only force acting upon the object is the Earth's gravitational force?  Relevant Information The following discussion seems to have solved exactly the same problem:   However, upon working out the mathematics, I'm not exactly sure how to evaluate the constant of integration.   A Partial Solution $$ F=\frac {-GMm}{s^2} $$ $$ a=\frac {-GM}{s^2} $$ $$ \frac {dv}{dt} = \frac {-GM}{s^2} $$ Multiplying by $v$ and then integrating by $dt$ on both sides, we have $$\frac {1}{2} v^2=\frac {GM}{s} +c_1$$ where $c_1$ is a constant of integration. Substituting initial conditions of $v=0, s=R$, we have $$\frac {1}{2} v^2=GM(\frac {1}{s}-\frac{1}{R})$$  At this point of time, when I use Wolfram Alpha, I get  $$c_2+\sqrt{\frac{2}{R}}t=\frac{\sqrt{s}(s-R)+R\sqrt{R-s}\times{\tan^{-1}(\sqrt{\frac{s}{R-s}})}}{\sqrt{GM(R-s)}}$$  where $c_2$ is a constant of integration. Substituting initial conditions of s=R, t=0, we find that the term $$\tan^{-1}(\sqrt{\frac{s}{R-s}})$$ is undefined. At this point, I'm stuck. Any ideas on where I've made the mistake here?  (For those interested, this question was inspired by the Greek myth which states that a bronze hammer dropped from heaven would take 9 days to hit the Earth and would reach on the tenth). $C \otimes A \cong C \otimes B$ does not imply $A \cong B$ Let $R$ be a commutative unital ring and let $M$ be an $R$-module and let $S$ be a multiplicative subset of $R$.  Today I proved both of the following: $$ S^{-1} R\otimes_R S^{-1}M \cong S^{-1} M$$ and $$ S^{-1} R \otimes M \cong S^{-1} M$$  Now I'm slightly confused.      Either my proofs are wrong or $C \otimes A \cong C \otimes B$ does not imply $A \cong B$. But I can't come up with an example. Can someone give me an example? (Or tell me that my proofs are wrong.) What is the C# equivalent of friend? Possible Duplicate:          I'd like the private member variables of a class to be accessible to a Tester class without exposing them to other classes.   In C++ I'd just declare the Tester class as a friend, how do I do this in C#? Can someone give me an example? I'm going to submit form using cURL, where some of the contents is come from other file, selected using sed  If param1 is line matching pattern from other file using sed, below command will works fine:  curl -d param1="$(sed -n '/matchpattern/p' file.txt)" -d param2=value2 http://example.com/submit   Now, go to the problem. I want show only text between 2 matching pattern excluding the matching pattern itself.  Lets say file.txt contains:  Bla bla bla firstmatch It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. secondmatch The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English.   Currently, lots of "beetween 2 matching pattern" sed command won't remove firstmatch and secondmatch.  I want the result to become:  It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. Is it possible to run Snow Leopard on a LATE 2011 MacBook Pro? I want to buy a new MacBook Pro, from Apple. However I want to run Snow Leopard instead of Lion.  I did some Google searches and found a few success stories of people who got SL to run on their early-2011 MBPs. I didn't find much of any experiences of people doing the same on a late-2011 MBP though, so I'm wondering (and really hoping!) if it's possible? Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$ $\lim_{x \rightarrow \infty} \lim_{n \rightarrow \infty} f_{n}(x) = \lim_{n \rightarrow \infty}\lim_{x \rightarrow \infty}f_{n}(x) $ in case of uniform convergence of $f_{n}(x)$ to $f(x)$  I used this while I was given that $f_{n}(x)$ converges to $f(x)$ uniformly. Is this correct and I was also thinking to prove this! Is the wave-particle duality a real duality? I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right? What does "Meta" mean? I am fairly new to Stack Overflow and just stumbled across Meta Stack Overflow.  I can't seem to figure out what the purpose of the "Meta" site is. Can someone please explain it to me? Is "has or will read" grammatical? I just wrote  in an ELL chat room. But looking at it (or more accurately, listening to my "inner voice" trying to "read it aloud"), I find it bothers me a lot.  A quick check on Google Books finds a claimed 432 written instances of , and 910 instances of , which suggests a significant minority of writers don't have a problem with the fact that the two different read's don't sound the same.  When I check the same construction with other verbs that don't have the same written form for past participle and present tense, it seems people nearly always include both (e.g. :1230, :8; :357, :7).  I'm not usually a big fan of "grammatical rules", but it seems to me there "ought" to be a rule that you shouldn't delete one instance of the verb unless it's "the same" as the one you're keeping. And it also seems to me that since language is primarily spoken, "the same" ought to mean "sounds the same when spoken", not "looks the same when written".  Can anyone who knows more than me about formal rules of grammar settle this one?    As an aside, offhand I can't think of any verb where the past participle and present tense sound the same but are written differently (maybe there aren't any), but would deletion be okay in that case? I'm making an python script which create a lot of object and this could take a lot of time. My question is how to show to the user of my script, the "progression" of the script? For now I print an progress bar in the system console.  But you have to open the system console befor and (apparently) the system console only work with Windows.  Is there an other way to show the progression of my script with python like control the mouse percentage icon (the little icon that replace the mouse while baking) or another idea? Is "until" inclusive or exclusive? I sent an email to someone and got an auto-reply saying:     I am out of the office until 09/15/2014.   Does this mean he will be available on the morning of the 15th? I was wondering if there is a simpler way to find the last to digits of a power such as $3^{45}$. I reduced it modulo 100 to get the answer, which is 43. But I was curious if there was a simpler, or more eloquent way to get this result.  Here's my method:  Using the fact that $3^4=81\equiv -19 \pmod{100}$ and $19^5=2,476,099\equiv -1 \pmod{100}$  $$ 3^{45}\equiv 3(3^4)^{11}\equiv 3(-19)^{11}\equiv 3(-19)(19^5)^2\equiv 3(-19)(-1)^2 \equiv -57 \equiv 43 \pmod {100} $$  I just felt like this was sort of a messy way to find out that the last two digits are 43. $F(x)$ versus $F[x]$ where $F$ is a field By 18 years of age, I want a brand new car that's $43,668 I'm from Arizona and I will be 18 in 4 months. I make about $1,200 a month from my job. I have $600 in my bank account and I don't plan to  spend any of it until I'm  18.     I'm looking to get a brand new Jeep Wrangler Rubicon that costs about  $43,000. The down payment is 10% (about $5,000) with an APR of  4.28%. The duration is 60 months with a monthly payment of  $800.  I don't have any credit. My friend suggested that I ask my parents to put my name on one of their credit cards so that I can build some credit. I'm not sure how that works. Can someone suggest a good way of going about this so I can get this car on or around my birthday?  My mom does finance work  for a big company and she told me that I should get a cheaper car. I really want this one and I'm willing to work for it. I'm getting a second job to get more money rolling in. My mom said that the only way that this would work is if I have a down payment of $15-20k. She also said that not everyone qualifies  for a car like this one. What can get me to qualify ASAP when I turn 18? I'm using an Android smartphone that I purchased Minecraft PE on; now I want to transfer my world to a iPhone. I have also purchased Minecraft PE on my iPhone. Is this sentence acceptable?     You’re welcome, have a nice day ahead. Combine multiple results in a subquery into a single comma-separated value I've got two tables:  TableA ------ ID, Name  TableB ------ ID, SomeColumn, TableA_ID (FK for TableA)   The relationship is one row of TableA - many of TableB.  Now, I want to see a result like this:  ID     Name      SomeColumn  1.     ABC       X, Y, Z (these are three different rows) 2.     MNO       R, S   This won't work (multiple results in a subquery):  SELECT ID,        Name,         (SELECT SomeColumn FROM TableB WHERE F_ID=TableA.ID) FROM TableA   This is a trivial problem if I do the processing on the client side. But this will mean I will have to run X queries on every page, where X is the number of results of TableA.   Note that I can't simply do a GROUP BY or something similar, as it will return multiple results for rows of TableA.   I'm not sure if a UDF, utilizing COALESCE or something similar might work? plist or sqlite If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.