If $A=\left( \begin{array}{ccc} 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \end{array} \right)$, find the rank of $A+A^2$  Note: Ans is given as Rank=2.   $A(e_1) = e_3, A(e_2) = e_1, A(e_3) = e_2$. Thus $(A+A^2)(e_1) = e_3+e_2$, $(A+A^2)(e_2) = e_1+e_3$ and $(A+A^2)(e_3) = e_2+e_1$. Thus $A+A^2$ has rank .....??????.
If $A=\left( \begin{array}{ccc} 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \end{array} \right)$, find the rank of $A+A^2+A^3$  We have $\rho(A+A^2+A^3)\leq\rho(A)+\rho(A^2)+\rho(A^3) $ where $\rho(A)$ means rank of A.  Here $\rho(A)=3.$ Also $\rho(A^2)=3$ and $\rho(A^3)=3$  as $\det A\neq 0$ and $\det(A^2)=\det(A)\det(A)\neq 0$ and so on.   But how to get the desired result.
I have the following image    that was created with this code:  \documentclass[varwidth=true, border=2pt]{standalone} \usepackage{tikz} \usepackage{tkz-euclide}  \begin{document} \usetkzobj{all} \begin{tikzpicture}     \tkzSetUpPoint[shape=circle,size=10,color=black,fill=black]     \tkzSetUpLine[line width=1]     \tkzDefPoints{0/0/A, 3/0/B', 2/2/C, 4/4/C'}     \tkzDefLine[parallel=through C](B',C') \tkzGetPoint{Phelper}     \tkzInterLL(A,B')(C,Phelper) \tkzGetPoint{B}     \tkzDrawLine[add=0 and 0.2](A,B')     \tkzDrawLine[add=0 and 0.2](A,C')     \tkzDrawSegment(B',C')      \node at ($(A)+(-0.1,-0.2)$)  {$A$};     \node at ($(B')+(0.2,-0.2)$)  {$B'$};     \node at ($(C')+(0,0.4)$)  {$C'$};     \node at ($(B)+(0.2,-0.2)$)  {$B$};     \node at ($(C)+(0.28,0.5)$)  {$C$};     \tkzDrawPolygon[ultra thick,color=blue,fill=blue!20](A,B',C')      \tkzDrawPolygon[line width=0.3pt,color=red,fill=red!20](A,B,C)      \tkzDrawPoints(A,B',C',B,C)     \tkzLabelSegment[below,red](A,B){$c$}     \tkzLabelSegment[left,red](A,C){$b$}     \tkzLabelSegment[right,red](B,C){$a$}     \tkzLabelSegment[below,blue,pos=0.8](A,B'){$c'$}     \tkzLabelSegment[left,blue,pos=0.8](A,C'){$b'$}     \tkzLabelSegment[right,blue](B',C'){$a'$} \end{tikzpicture} \end{document}   As you might notice, the space above C' is bigger than necessary. This is because of Phelper. I need Phelper for the construction, but it should not influence the image size in any way. I've tried to get Phelper closer to C, but that seems not to be possible.  How can I stop defined (but not visible) points from influencing image size?  Please note: I wrap code like this in a figure. So images like the one above are only a part of the whole document. So working with standalone does not solve my problem. Every irreducible Polynomial is primitive or irreducible constant Proof of the uncountability of reals using the diagonal argument—problem? I keep getting asked to do wedding photography - what is a fair price to ask? This may sound like a silly post...but, I need some "outside" opinion from people who know what they are doing.  I took up amateur photography as a hobby a couple of years ago...most of it being portrait photography of my children.  As I have posted my pictures on facebook, a plethora of people have loved my "work" and have been contacting me about my "photography" business.  I have been very up-front and honest with every one of them about only being an amateur, and never professionally trained.  That being said, people still hound me to take their pictures.  I am comfortable with my ability to take portrait style pictures....and I have even started charging $60 for a two hour shoot with the rights to the cd of prints.  I do not do my own printing.    Now, I have done three weddings for family members just over this last summer....and only as a wedding gift to them (in other words, I have not charged.)  But now, I am having others, who are not family members, wanting to "hire" me for their weddings.  Again....I have tried to tell them I don't do professional photography, only to be met with all of them still wanting to hire me.    My question is...what is a fair price to ask of these people?  They obviously still want me to work for them, knowing that I have no credentials.....but I have no idea how or what to charge for.  Can anyone give me some ideas? Inverse elements in the absence of identities/associativity. Show that one-sided limits always exist for a monotone function (on an interval) Show that one-sided limits always exist for a monotone function on an interval $[a,b]$.  Me attempt:  1) If a function is monotone on an interval $[a,b]$, then $f(a)\le f(x) \le f(b)$ for $x\in[a,b]$. Therefore if there exists left-hand (right-hand) limit of this function at a given point, then it must be finite. Now we must show that a both left-hand and right=hand limits exist. We use the fact that if we take a sequence $(x_n)\rightarrow c^{+} \in [a,b]$ then $f(x_n)$ is bounded and monotone and therefore it is convergent. How to write the word 'smart phone' in a research article? Smart phone Smartphone Smart-phone   I am really looking for some help regarding hyphen (-) issue in text found in research article. Is there any standard method how to use it? Is it possible for an American to get a Chinese visa in Hong Kong? In the PHP manual, to show the syntax for functions with optional parameters, they use brackets around each set of dependent optional parameter. For example, for the  function, the manual reads:  string date ( string $format [, int $timestamp = time() ] )   Where $timestamp is an optional parameter, and when left blank it defaults to the  function's return value.  How do you go about creating optional parameters like this when defining a custom function in PHP? I have two sites that were originally migrated from SPS2003.  It worked well in SharePoint 2007 and even on SharePoint 2010,  however when we migrated it to SharePoint 2013,  the home page throws following errors on the ULS Log.     5/24/2013 09:57:46.62  w3wp.exe (0x0D50)   0x1D24  SharePoint Foundation   Files   ak8dj   High   UserAgent not available, file operations may not be optimized.    at Microsoft.SharePoint.SPFileStreamManager.CreateCobaltStreamContainer(SPFileStreamStore spfs, ILockBytes ilb, Boolean copyOnFirstWrite, Boolean disposeIlb)     at Microsoft.SharePoint.SPFileStreamManager.SetInputLockBytes(SPFileInfo&amp; fileInfo, SqlSession session, PrefetchResult prefetchResult)        05/24/2013 09:57:46.62 w3wp.exe (0x0D50)   0x1D24  SharePoint Foundation   Cobalt  ajoii   Medium   Mapping Store error to Cobalt error. [Store Error: System.NullReferenceException: Object reference not set to an instance of an object.     at Microsoft.SharePoint.CoordinatedStreamBuffer.SPCoordinatedStreamBufferFactory.CreateFromDocParams(SqlSession session, Guid guidSiteId, Int32 cbContent, Nullable1 parentId, Guid docId, Nullable1 level, SPChunkedArray`1 rgbContent, Byte[] rgbRbsId, Int64 bsn, Byte partition, Int32 pageSize, Boolean bStartFilling)        05/24/2013 09:57:46.62 w3wp.exe (0x0D50)   0x1D24  SharePoint Foundation   Cobalt  aintx   High    Failed in CobaltStream.LockBytes: ErrorException --- Error: Unknown (Win32=0x80004003) --- Error message: System.NullReferenceException: Object reference not set to an instance of an object.     at Microsoft.SharePoint.CoordinatedStreamBuffer.SPCoordinatedStreamBufferFactory.CreateFromDocParams(SqlSession session, Guid guidSiteId, Int32 cbContent, Nullable1 parentId, Guid docId, Nullable1 level, SPChunkedArray`1 rgbContent, Byte[] rgbRbsId, Int64 bsn, Byte partition, Int32 pageSize, Boolean bStartFilling)     at         05/24/2013 09:57:46.64 w3wp.exe (0x0D50)   0x1D24  SharePoint Foundation   Files   ak8dj   High    UserAgent not available, file operations may not be optimized.    at Microsoft.SharePoint.SPFileStreamManager.CreateCobaltStreamContainer(SPFileStreamStore spfs, ILockBytes ilb, Boolean copyOnFirstWrite, Boolean disposeIlb)     at Microsoft.SharePoint.SPFileStreamManager.SetInputLockBytes(SPFileInfo&amp; fileInfo, SqlSession session, PrefetchResult prefetchResult)     at         05/24/2013 09:57:46.64 w3wp.exe (0x0D50)   0x1D24  SharePoint Foundation   Database    ab1a9   High    Failed to get document content data. System.NullReferenceException: Object reference not set to an instance of an object.     at Microsoft.SharePoint.CoordinatedStreamBuffer.SPCoordinatedStreamBufferFactory.CreateFromDocParams(SqlSession session, Guid guidSiteId, Int32 cbContent, Nullable1 parentId, Guid docId, Nullable1 level, SPChunkedArray`1 rgbContent, Byte[] rgbRbsId, Int64 bsn, Byte partition, Int32 pageSize, Boolean bStartFilling)     at   I could not find any materials related to this on the internet,  I am not even sure if it has to do something with SPS2003 compatibility.  Any help pointers? Under what conditions is $AA^T$ invertible? Given a matrix $A$ with dimensions $m \times n$, is $B=AA^T$ invertible if and only if the rows of $A$ are linearly independent?  So far, I've tried writing A as row vectors,   $$A = \begin{bmatrix}v_1\\ v_2\\ \vdots \\ v_m\end{bmatrix}$$  where $B_{i,j} = (v_i v_j) $  The Wikipedia page on the  suggests that the condition I stated above is necessary and sufficient, but does not provide a proof. How can I get suffix matching in a search in SharePoint 2013? I'm new to SharePoint 2013 and I'm quite puzzled, that it seems to be impossible to search for the middle part of a string.  Simple example:  I've added a lot of documents (PDFs, ...) from our Windows Server to the SharePoint Library and I want to add metadata. I try to filter the view of the document library, to only get those documents which I currently want to edit.  The names of the documents are very systematic, composed of naming conventions with abbreviations like:  Superthing2000-b_Suppl_REF_13580_note_1108_04en which means e. g.: Document about Product "Superthing2000", variant "b", Supplement to the manual, ID "13580", from 08/2011 in english language.  Now, I'd like for example to filter out all documents containing ID 13580 or "en" for english, but no chance, as suffix matching is not possible, even if I explicitly search for e. g. *en*.  Is there a way to define a search with kind of a "contains" operator which also finds files when the search string is only a part in the middle of the file name? Even the Windows Explorer search feature is able to do that!! swap x and y axis on data generated plot My adoptable storage is full. How can I move the data to a bigger MicroSD card? I've got a Moto G 2014 LTE running CM13. When I found out about the whole 'adopt SD card as internal storage' thing, I was quite excited, as the actual internal storage on this phone is tiny. I set up a 16gb micro SD, the biggest I had at the time, and have been using it with minimal problems.  Fast-forward to this day, and I now have a 64gb micro SD that I want to use instead of the 16gb one. However, I'm at a loss as to the best way to copy the data over onto the new one, or if it's even possible. The ideas I've got so far include:   Titanium Backup all [system] apps, factory reset, format new SD and then restore (should work, but a massive pain) NANDroid the SD card onto a USB OTG stick, then restore to the new one (pretty sure this is a bad idea and won't work, but...) Copy all the stuff onto my computer via my phone through a file manager of some sort (I have a linux/linux/windows triple boot, so file format isn't a problem), format the new SD into phone, and copy back (can't be that simple, can it? Hidden/inaccessible files?) Copy the files onto USB OTG with a root file explorer app, reformat the SD as portable, format the new one as adopted, and then copy them back (could work, except for the point below)  with the command adb shell sm set-force-adoptable true, use a root browser to copy all the data over from the old SD, format the old SD as external and remove it, and take the new SD and put it in the SD card slot (seems overly complicated, and also suffers the below pitfall)   The other thing that worries me is mount points: will the apps with externally-stored data know to look at the new SD card for it? Will I have to remount it in a specific location with the shell each boot (or use an init.d script perhaps)? I suppose that it can always be accessed through /sdcard or /storage/emulated/0, but I'm not 100% sure what goes on behind the scenes.  I've got recent NANDroid backups, so if anyone can eliminate the options that definitely won't work (or say which one[s] probably will), I can try them and report back in case someone stumbles across this question in the future.  EDIT: So, my phone further bit the dust and got to the point where it got into a pretty major bootloop. Ended up having to flash a whole new ROM. As such, I probably won't be able to test the full image/decryption/etc. method that I was originally planning. However, I will write up what I've discovered so far in case anyone wants to take up this particular method in the future:   Image the old SD with dd as follows: sudo dd if=/dev/mmcblk0 of=~/SDCardBackup.img, where /dev/mmcblk0 is the location of the sd card as obtainable through lsblk Restore the image you just made to the NEW sd card : sudo dd bs=4M if=~/SDCardBackup.img of=/dev/mmcblk0 (you may have to unmount it first) From here on in, I was following  (thanks @beeshyams!) tutorial, which explains the next steps way better than I ever could have done. The plan was to decrypt the storage and resize the main partition with gparted. I had no idea whether it would work or not, but I felt it was worth a try. However, when I came to run the actual decryption command using the key I'd extracted, I got an error (which I believe was my shoddy linux skills more than anything).   As I say, I have no idea if the above method would work or not even if everything did go to plan. Also, feel free to correct any of my commands above if I mis-typed them (I'm on Windows currently, so this is all from memory.) If this info helps anyone, then good. But until then,  should work fine.  Thanks, Jackdafish First: I'm obviously not talking about logging in and then automatically launching Terminal.app.  In , you used to be able to type &gt;console in the username field on the login screen and go directly into a command line terminal session.  I haven't done it in years until the other day. However, now, when I try it, my computer just locks up. I can't cancel, and it never sends me to a terminal session.  Is it even possible to do this anymore? If not, does anybody know when it was removed? It seems odd that typing &gt;console would cause the machine to lock up if it wasn't recognizing it in some way. help with some Hilbert style proofs in a propositional logic axiom system. I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. $\lim_{x\to 0} \frac{\sin x - x}{x^2}$ without L'Hospital or Taylor Why is it sensical for a proposition with a false antecedent to validate to true? What do I need in a camera to take non-blurry photos of a fast-moving child? Should I use a comma in this sentence? I am trying to teach myself punctuation rules. I have this sentence:     The Disney version generally considered both racist and sexist portrayed the female characters in stereotypical roles and presented ethnically diverse characters with stereotypical features.   I think the sentence needs a comma after sexist and before portrayed. I also think it needs a comma after roles because of what I have identified as a coordinating conjunction. I.e.:     The Disney version generally considered both racist and sexist, portrayed the female characters in stereotypical roles, and presented ethnically diverse characters with stereotypical features.   Did I punctuate the sentence correctly? Is this a coordinating conjunction?