Eigenvectors of a Hermitian operator corresponding to different eigenvalues are orthogonal. Even for a degenerate eigenvalue we can produce orthogonal eigenvectors in that eigensubspace. Does this system of orthogonal vectors necessarily span the whole vector space, i.e., do they constitute a basis?
I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of another operator $L_z$. I understand that I have to "expand" the state vector in the $L_z$ basis, which in this case is just the appropriate eigenvector, and then change the base to that of $L_x$, i.e to write the $L_z$ eigenstate in the $L_x$ base. The problem is that there is just no way of doing it, since the eigenvectors of $L_x$ don't span the whole Hilbert Space.   I thought all quantum operators had a set of eigenvectors that spanned the whole space, am I wrong on that or am I wrong on my algebra?
Autodesk Maya symbol lookup error on start after installing AMD proprietary driver on Ubuntu 18.04 I've installed Maya 2018 on Ubuntu 18.04 and tested it with the amdgpu driver (open source) and it launched without problems, but some components were not working with the open source driver so I tried amd gpu pro 18.20 (amd closed-source driver). I ran  ./amdgpu-pro-install   but since then Maya won't work and gives this warning message:  ./maya.bin: symbol lookup error: /usr/autodesk/maya2018/bin/.. /lib/libOGSDeviceOGL4-16.so: undefined symbol: EVP_CIPHER_CTX_init   My card is hd7770 and it's supported by the amd closed driver, so what am I doing wrong? GNU screen: Running commands in different window when screen is started What is the name of this kind of character illustration? I have seen lots of UI designer nowadays including this kind of illustration i want to know the name of this style so that i can also learn and try this in my designs. I have an internet connection that I'd like to be enabled only during the day. With multiple household internet devices, this would need to be something disabled at the level of the router.  I don't believe the router offers this sort of functionality, but I'm reasonably familiar with setting up web servers and the like, and I'm wondering if I can set up some sort of proxy server on the internet that the router must go through and have that proxy only active at certain times of day.  It's a bit vague I know, but is something like this even possible?  If not, are there other solutions to the problem? I am being flown out by a company for an on-site visit. Will I earn miles for this trip, or will the company? How to remove Ubuntu and put Windows back on? How do I remap the keys on my keyboard? I broke my laptop and had to buy a new one with a new keyboard that I am struggling to get familiar with, especially with the keys like home, page-up, page-down and end, which aren't standardized at all.  I found myself doing this for like the 5th time in the last several years, and I think it is really annoying.   Is there an app that can reliably map one key to another?  By reliably I mean that key mapping works with Shift, Alt, and other modifiers.  UPDATE  Windows 7, Samsung Chronos 7 How to compute $m$ value from RSA if $phi(n)$ is not relative prime with the $e$? Here is some information we got : We know the value of $n$, with size $1043$. We know the value of $p$ (size $20$) and $q$ (size $1023$) as the factors. $e = 65537.$ $\varphi(n)$ = $(q-1)(p-1)$ When I calculated $\gcd$ and $\text{modinv}$, I got : $\gcd(e,\varphi(n)) = 65537$ $modinv(e,\varphi(n)) = 1 $ So we can tell that they are not relatively prime. So, how to compute the d, and get the value of m? I'm not that good with math, so I cant understanding well the theory. so can anyone please make an example implementation or write a clear formula? I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.  I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.  I want to see an actual proof behind this equation.  Does anyone have any ideas? OpenSSL and error in reading openssl.conf file What if table columns cannot fit on webpage? If I've 12 - 14 columns in a table to show on a page but there's not enough space to show all columns (let's say, there's only enough space to show 7 columns of the table), what is the solution for this?  I can make the page scroll horizontally but it's ugly and the last resort. Shrinking the font size of the table data only has rooms for 1  - 2 more columns at best. Add leading zeroes to number in Java? Is there a better way of getting this result? This function fails if num has more digits than digits, and I feel like it should be in the library somewhere (like Integer.toString(x,"%3d") or something)  static String intToString(int num, int digits) {     StringBuffer s = new StringBuffer(digits);     int zeroes = digits - (int) (Math.log(num) / Math.log(10)) - 1;      for (int i = 0; i &lt; zeroes; i++) {         s.append(0);     }     return s.append(num).toString(); } First part :  I want to prove the following De Morgan's law :  ref.(dfeuer)  $A\setminus (B \cap C) = (A\setminus B) \cup (A\setminus C) $    Second part:  Prove that $(A\setminus B) \cup (A\setminus C) = A\setminus (B \cap C) $    Proof:  Let $y\in (A\setminus B) \cup (A\setminus C)$     $(A\setminus B) \cup (A\setminus C) = (y \in A\;  \land y \not\in  B\;) \vee (y \in A\; \land y \not\in C\;)$      $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= y \in A\ \land (  y \not\in B\; \vee \; y \not\in C ) $      $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= y \in A \land  (\lnot ( y\in B) \lor \lnot( y\in C) )$   According to De-Morgan's theorem :  $\lnot( B \land C) \Longleftrightarrow (\lnot B \lor \lnot C)$ thus     $y \in A \land (\lnot ( y\in B) \lor \lnot( y\in C) ) = y \in A \land y \not\in (B \land C)$   We can conclude that  $(A\setminus B) \cup (A\setminus C) = A\setminus (B \cap C)$ How to get the current time in Python What is the module/method used to get the current time? Why is this clause unsatisfiable? Add folder to PATH How to add something to PATH ? I have added like   PATH=$PATH:/home/example export PATH   and it works until I restart , after that when  I type env in PATH there is no /home/example. Did Sidious know the secret of cheating death? How do I control how Emacs makes backup files? Emacs puts backup files named foo~ everywhere and I don't like having to remember to delete them.  Also, if I edit a file that has a hard link somewhere else in the file system, the hard link points to the backup when I'm done editing, and that's confusing and awful.  How can I either eliminate these backup files, or have them go somewhere other than the same directory? Oddly wide spaces in subsection title I have a problem, this is what it looks like:     And this is the code:  \subsection{Bakterienst\"amme und Anzuchtbedingungen}   Where does the huge whitespace come from? Show that  the  alternating  group  $A_4$  of all even permutations of $S_4$ does not contain a  subgroup of order $6$.   For me am thinking to write all elements of $A_4$ and trying to find every cyclic subgroup generated by each element of $A_4$, then I have to check whether there exist such a subgroup or not! This is a long procedure for me, I ask if there is a short way to do this.