help with linear algebra proof
linear dependence proof using subsets
There's something wrong with the vertical alignment on the :   (click to enlarge)  On the , which uses a similar list but without the Favorite column, the vertical alignment is correct.  (Tested in the latest Firefox, Safari and Chrome on macOS Catalina.) "Was going to" versus "would be" as the past tense of "will" Do you think that "was going to be" or "would be" could be used as past tense of will?     I thought this book was going to be your choice.        I thought this book would be your choice.   Do these two sentences basically mean the same thing or are they slightly different? What do you call snow that has collected between a car body and the tire? Does non-mass-energy generate a gravitational field? At a very basic level I know that gravity isn't generated by mass but rather the  and when I wave my hands a lot it seems like that implies that energy in $E^2 = (pc)^2 + (mc^2)^2$ is the source of gravity.  If the total energy of a particle contributes to the generation of a gravitational field, that seems to imply some strange things.  For example, you could accelerate a particle like an electron to such a high velocity that it has enough momentum / kinetic energy to collapse into a black hole.  An observer traveling with the electron would see an electron and stationary observers would see an arbitrarily massive black hole flying by.  These observations seem to be contradictory.  The same would be true of an extremely high energy photon.  Another consequence is that a stationary particle could appear to have so much mass that it could attract / deflect the trajectory of a rapidly moving observer.  It seems odd that at low velocities you wouldn't even notice a stationary electron as you pass it by but at high velocities the gravitational field you observe could be significant enough to alter your path.  Am I correct that non-mass-energy generates a gravitational field?  If it does, how are these strange observations like a particle looking like a black hole in one reference frame but not another reconciled? How to do joins in LINQ on multiple fields in single join I need to do a LINQ2DataSet query that does a join on more than one field (as  var result = from x in entity join y in entity2         on x.field1 = y.field1  and            x.field2 = y.field2   I have yet found a suitable solution (I can add the extra constraints to a where clause, but this is far from a suitable solution, or use  solution, but that assumes an equijoin).  Is it possible in LINQ to join on multiple fields in a single join?  EDIT  var result = from x in entity              join y in entity2              on new { x.field1, x.field2 } equals new { y.field1, y.field2 }   is the solution I referenced as assuming an equijoin above.  Further EDIT  To answer criticism that my original example was an equijoin, I do acknowledge that, My current requirement is for an equijoin and I have already employed the solution I referenced above.  I am, however, trying to understand what possibilities and best practices I have / should employ with LINQ. I am going to need to do a Date range query join with a table ID soon, and was just pre-empting that issue, It looks like I shall have to add the date range in the where clause.  Thanks, as always, for all suggestions and comments given How far can I refactor the code in someone else's question? How do I put big brackets under different parts of an equation to attach text labels? How do I put big brackets under different parts of an equation so that I can write, e.g., text specific to these parts? For example: f =  x^3 +  2     |___|  |__|        |     |    text 1  text 2 What triggers a full garbage collection in Java? Given:  DateTime.UtcNow   How do I get a string which represents the same value in an -compliant format?  Note that ISO 8601 defines a number of similar formats. The specific format I am looking for is:  yyyy-MM-ddTHH:mm:ssZ I pressed F2 to take a few Minecraft screenshots. Later I couldn't find them. Where are they stored on my computer? What can be considered as "Home ties" for a UK Standard Visitor visa (PLAB)? Why are polymorphic values not inferred in Haskell? Question:  Show that $n^2 + 3n + 5$ is not divisible by $121$, where $n$ is an integer. Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$ My installer build "signs" a DLL using a Code Signing certificate during the build process.   I've noticed that if I try to build twice in succession, the second build fails because the DLL is already signed so signcode chokes. Obviously I can fix this by signing a copy of the DLL in the build, but the problem intrigued me:   Is it possible to "unsign" a DLL, and if not, why not...? Why does C++ not let baseclasses implement a derived class' inherited interface? What is the /boot partition really for? I'm reading a relatively old text on Linux partitions and file systems (the LPIC 1 Certification Bible). It says:     Some versions of the Linux boot loaders cannot access a kernel that is   outside the first 1024 cylinders on a disk. By putting the /boot   partition at the beginning of the drive you can be assured of not   having a problem when accessing the kernel at boot. This problem shows   itself most often in cases of dual booting Linux along with another   operating system that is on the first partition.   Why would a boot loader have "no access to the kernel outside the first 1024 cylinders on a disk"?  Also, what does "putting the /boot partition at the beginning of the drive" mean? I have two atoms, both of which are either bosons or fermions, with four allowed energy states: $E_1 = 0$, $E_2 = E$, $E_3 = 2E$, with degeneracies 1, 1, 2 respectively.  What's the difference between the partition functions of a pair of two bosons and that of a pair of two fermions? Proving Pascal's identity So I came across Pascal's identity: Prove that for any fixed $r\geq 1$, and all $n\geq r$, $$ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}. $$  I know you can use basic algebra or even an inductive proof to prove this identity, but that seems really cumbersome. I was wondering if anyone had a "cleaner" or more elegant way of proving it. For example, I think the following would be a decent combinatorial proof.   Proof: Let $S$ be a set with $n+1$ elements, and consider some fixed $x\in S$. There are $\binom{n+1}{r}\;\; r$-subsets of $S$--count them according to whether or not they contain $x$: there are $\binom{n}{r}$ not containing $x$, (each formed by choosing $r$ of the remaining $n$ elements in $S\setminus\{x\}$), and there are $\binom{n}{r-1}\;\; r$-sets containing $x$, (each formed by selecting an additional $r-1$ elements in $S\setminus\{x\}$).   Is that right? Are there any other efficient ways of doing it? How can I prevent URLDownloadToFile from retrieving from the cache? I am using URLDownloadToFile to retrieve a file from a website. Subsequent calls return the original file rather than an updated version.  I assume it is retrieving a cached version.