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Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $ Where $ f: V \rightarrow V  $ is a linear operator on an inner product space $V$, I'm trying to show that $$ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $$  i.e. the Kernel of the adjoint is equal to the orthogonal complement of the image of f.  I taking the inner product of $f(v)$, where $v \in V$  and an arbitary element of the kernel, $k \in \mathrm{ker}(f^*) $  $$\langle f(v), k\rangle = \langle v, f^*(k)\rangle = \langle v, 0\rangle = 0$$  And then taking the inner product of $f(v)$ with $w$, where $w \in  \mathrm{im}(f)^{\perp}$  By definition  of the orthogonal complement to the image of f,   $$ \langle f(v), w\rangle = 0 $$  My question is, now that we have   $$ \langle f(v), k\rangle = \langle f(v), w\rangle = 0 $$  What can we conclude? I would like to conclude that because $w$ and $k$ are arbitary elements of their respective subspaces, that these must be equal, i.e. $ \mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $  However, I am not sure this is mathematically sound, is that conclusion valid or are there any technical barriers or additional statements that would have to be made to make that watertight? Probability of flipping coin What is the probability of obtaining THT before HHH when a toss is tossed sequentially? Could someone solve using state diagram and law of total probability? How do I safely delete all files with a specific extension (e.g. .bak) from current directory and all subfolders using one command-line? Simply, I'm afraid to use rm since I used it wrong once and now I need advice. Scaling in object and edit mode My question is: why do Object mode scaling and Edit mode scaling seem to be incompatible? I'm quite new to blender, so I may not know sth important about these modes. The actual case is:  I create a cube, which is 2×2×2 metres in metric units by default  (it's also shown under 'dimensions')    Then, with the cube selected, I switch to Edit mode, and enable 'Edge info: Length' under 'Mesh display'    And it shows the cube's measurements, which are 2m on each edge.  Now I use the scale tool in Edit mode, making the edges 2.244m    That's good so far. Changing back to Object mode, the changes can be seen under 'dimension'    However, when I use the scale tool in Object mode, the scale and dimensions also change, but changing back to Edit mode, the measurements shown remain the same      So the question is, why does the scale tool work differently and have different effects in Ojbect mode and Edit mode? Also, how can I enter the desired size of my cube manually to get an accurate result, without using the scale tool in Edit mode, and without changing the scale ratio in object mode? How to reverse name and first name? With biblatex and verbose-trad2 style, I would like to reverse the order of the last name and the first name for inline references.  The default behavior is:     John DOE, Title, place : publisher, year   for inline references (in my case, in a footnote with footcite command) and:     DOE, John, Title, place : publisher, year   in the bibliography at the end of the document.  I would like to have both inline and final references in the following format (similar to the bibiography style).      DOE, John, Title, place : publisher, year   P-S : There are some close questions (like ) but only about the final bibliography (and no inline references). \DeclareNameAlias{sortname}{last-first} works only for the ending references (not inline ones). Audits bug in the filtered review queue Just recently the . I've got  this morning which seems to be out of scope for my filtered list. I have filtered my list to only review VBA and Excel-VBA tags while the audit is tagged with perl http pipes which have nothing in common with VBA. I see a problem with the way the system currently works  Since I have a filtered list of tags, once I see a differently tagged question I already know it's an audit. Come on, that's too easy!  Can this please be fixed? update I have ran into  which I decided to edit - the question was tagged VBA and I thought it had nothing to do with VBA so I selected Edit which was the correct action. Now, once the first audit was clearly one to close - nothing to do with VBA and really easy to know that it actually was an audit - the second one was a very good audit. I am just thinking now if this is a proper bug or just a matter of malfunction in the system. I think the first one may have been made an audit for the wrong reason. Let's turn this into a discussion as I can't clearly work the system out. What does a "CALLBACK" declaration in C do? I was looking through some code from the  library and came across a function declared like this:  LRESULT CALLBACK WndProc(HWND hwnd, UINT msg, WPARAM wParam, LPARAM lParam)   Now, I'm a Delphi coder.  No hablo C muy bien, senor.  But I remember enough syntax from my college courses to read it like this:  Function name is WndProc.  Argument list is pretty self-explanatory.  Function return type is LRESULT.  But what in the world is that "CALLBACK" doing there?  In Delphi, any function can be used as a callback; you just need to pass the right type of function pointer.  Is there any particular reason why C doesn't work that way?  Or does it mean something different? In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality? In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$  Alternatively, we can leverage the notion of a partial surjection, by which I mean a partial function that hits every point of its codomain at least once.  $$Y \succsim X \leftrightarrow \mbox{ there exists a partial surjection } Y \rightarrow X.$$  Now in ZFC, these approaches are equivalent, in the sense that $$X \lesssim Y \iff Y \succsim X.$$  But if we just consider ZF (that is, we drop choice) then the equivalence breaks down. However, we've still got the forward implication, because the converse of an injection is always a partial surjection.  A specific question. Define that two sets are weakly equipotent iff $X \succsim Y$ and $Y \succsim X$. Question: does weak equipotence imply genuine equipotence? I'm guessing "no."  A general question: In broad terms, how would the structure of the cardinal numbers in ZF be altered by adopting $\succsim$ rather than $\lesssim$ as the fundamental order relation, and taking equality of cardinal numbers to correspond to weak equipotence of sets? I'm guessing this would hide a lot of the (usually very complicated) structure of the cardinal numbers in ZF.  Remark. The order symbols can be written \lesssim and \succsim. Relation Between Bayesian Estimation and Maximum a posteriori estimation Is maximum a posteriori estimation some kind of Bayesian Estimation?   If yes, can you point out other Bayesian estimators?  Edit:  So I've come to know the following (don't know if they are correct):  MAP finds the mode of the posterior, and Bayesian Estimator finds the distribution of that posterior with respect to the estimating parameters. And in order for Bayesian Estimator to find the optimal parameters, some particular risk function (like Mean Squared Error, 0/1 error etc.) must be defined.  So it seems these two estimators have nothing in common except they both incorporate prior into their estimation. Or is it possible that MAP is some special case of Bayesian Estimator with some particular risk function. Enumerate paragraphs and subparagraphs like sections but with letters Running an infinite loop on startup I'd like to run a command at startup which pings a certain address every 10 minutes and writes the result to a file. I've figured out now how to do the pinging and file writing and the 10 min intervals:  while true; do my-command-here; sleep 600; done   My question is, can I put this in /etc/init.d/rc.local or should I be putting it in /etc/rc.local or somewhere else entirely? I'm specifically concerned because it's an infinite loop so I'm not sure if I could put it in one of these startup scripts.  Some help would be appreciated. I'm using Ubuntu 12.04.5 A generalized version of the second Borel-Cantelli lemma says     Theorem 5.3.2. Second Borel-Cantelli lemma, II. Let $\mathcal F_n, n \ge 0$ be a filtration   with $F_0 = \{\emptyset, \Omega\}$ and $A_n , n \ge 1$ a sequence of events with $A_n ∈ \mathcal F_n$ . Then   $$ \{A_n \,i.o.\} = \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\}. $$   An exercise of this lemma is     Exercise 5.3.6. Show    $\sum_{n \ge 2} \mathbb P (A_n | \cap_{m=1}^{n−1} A_m^c ) = \infty$ implies $P (∩_{m \ge 1} A_m^c ) = 0$.   I think we can actually prove that $\cap_{m \ge 1} A_m^c = \emptyset$. My proof is like this: Let $B_n = \cup_{m = 1}^n A_n$. Let $\mathcal F_n = \sigma(A_1,\dots,A_n)$, which forms a filtration. It is easy to verify that $$ \mathbb P (A_n | B_{n-1}^c ) = \mathbb P (A_n | \mathcal F_{n-1})(\omega) $$ when $\omega \in B_{n-1}^c$.  Assume there exist $\omega \in \cap_{m \ge 1} A_m^c$. It follows from the previous lemma that $$ \omega \in \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\} = \{A_n \,i.o.\}. $$ In other words, $\omega \in \cap_{m \ge 1} \cup_{n \ge m} A_n$, which contradicts $\omega \in \cap_{m \ge 1} A_m^c$.  Is this the right proof? How to create animated GIF images of a screencast? I've seen animated GIF images of screen casts (like the one below) promoted a few times on this site as a way to improve answers.    What toolchain is being used to create these? Is there a program that does this automagically, or are people taking screencasts, converting them into a series of static frames, and then creating the GIF images? How can i make the grass bend when this cube hits it?  Here's the picture: Image Alpha Background Renders Black when using "Import Images as Planes" I must be missing something here.  I'm doing an extremely simple Import Image as Plane using the AddOn and the imported image has an alpha background but it renders (and even previews in the material preview) as black where the alpha background should be.  When I import, I select Emission &amp; Transparent so the material setup is automatic.  I've read similar issues but people are typically missing the transparent node and connecting the Alpha channel from the image into the Mix Node factor.  But this is all here.  I must be missing something really stupid but I've wasted so much time trying to fix this thing.  Any help would be greatly appreciated.  Not sure if it matters but I'm using 2.74 RC2.  Many thanks!  I created this dummy blend file to test outside my project but problem still exists.  Here is the simple .blend file: how to install packages with apt-get on a system connected via proxy? I've got an Ubuntu 11.10 system connected via a proxy to the internet, and I would like to install packages via apt-get. When I try to do that I get this error message:  sudo apt-get install libboost-program-options-dev [...] Err http://gb.archive.ubuntu.com/ubuntu/ oneiric/main libboost1.46-dev i386 1.46.1-5ubuntu2 407  Proxy Authentication Required   Any ideas? How can I modify the advanced search form when my hook runs too early? Order of operations using Object Initializer Syntax Does the order in which I set properties using the object initializer syntax get executed in the exact same order?  For instance if I do this:  var s = new Person { FirstName = "Micah",                      LastName = "Martin",                      IsLoaded = true                    }   will each property get set in the same order? Finding files by only knowing extension? I've started using the dashicon fonts in my theme options page for WP 3.8+ users. To take care of users on &lt; 3.8, I've added the dashicon fonts to my theme folder and also an entry at the top of my custom admin stylesheet:  @font-face {     font-family: 'dashicons';     src: url('fonts/dashicons.eot'); }  @font-face {         font-family: 'dashicons';         src: url(data:application/x-font-woff;charset=utf-8;base64,xxx) format('woff'),         url('fonts/dashicons.ttf') format('truetype'),         url('fonts/dashicons.svg#dashicons') format('svg');     font-weight: normal;     font-style: normal; }   (I've left out the base64 encode for brevity)  This works great, however, I'm getting a bit of strange behavior with the fonts. When I leave the admin open in a tab and go away for a bit, then come back, every once in a while, the webfonts have turned to squares, as if the font is missing.  This does not happen every time. It appears to be somewhat random.  In those cases, simply doing a mouseover the page usually returns the webfonts to their proper symbols.  The screenshots below show the issue. The first screenshot is what I see when the problem occurs. The second screenshot is what I'm expecting to see when things go well.   Notice in the first shot, how the wordpress labels themselves have gone to a default serif font.