Is it the case that $A \times (B \times C) = (A \times B) \times C$ if and only if one of the factors is the empty set? I am talking about strict equality, not isomorphism. The duplicate question did not answer my question, because it never said when the associative law is true.
I have a basic doubt about the associativity of the cartesian product. Well, first wikipedia says that the cartesian product isn't associative, and there's a good argument for it: if $x\in E$, $y\in F$ and $z \in G$ the identity $((x,y), z)=(x,(y,z))$ would imply that $(x,y) =x$ and $(y,z) = z$ so that $((x,y),z)=(x,y,z)$ would mean nothing.  That's fine, I like this argument. However, in his book Calculus on Manifolds, Spivak says that the cartesian product is associative. He says: if $A \subset \mathbb{R}^m$, $B\subset \mathbb{R}^n$ and $C \subset \mathbb{R}^p$ then $(A\times B)\times C = A \times (B \times C)$, and both of these are denoted simply by $A \times B \times C$.  Well, this confuses me because Spivak is always very rigorous, so that he wouldn't state something that's not true in such way. Is Spivak or Wikipedia right ? Or Spivak's statement only works for subsets of euclidean spaces ?  Thanks in advance and sorry if this question is too basic.
Matrix of Linear Transformation by right multiplication I am trying to solve the following problem:   Let $A$ be an $n\times n$ matrix, and let $V$ denote the space of $n$-dimensional row vectors. What is the matrix of the linear operator ‘‘right multiplication by $A$’’ with respect to the standard basis of $V$?  I am not sure where to begin with this problem. Character reading from file in Python How to set a short password on Ubuntu? There are three random variables, $x,y,z$. The three correlations between the three variables are the same. That is,  $$\rho=\textrm{cor}(x,y)=\textrm{cor}(x,z)=\textrm{cor}(y,z)$$  What is the tightest bound you can give for $\rho$? Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$. Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$.  a) Find the base and dimension of fundamental subspaces of matrix A  b) Using the results given by a) find the general solution for $Ax - z_1$.  Any hint would be appreciated. Few days back while attending a thesis defense, one Professor was asking the defender why does he think, he deserves the degree. I was wondering since then, is there really any general answer for that? or the answer should be given describing my own work and then emphasis them how it is important for research field?  I asked the question here to know your opinion about it and to list down what could be the possible answers to this question. "Would" & " Used to" How to specify that search should be between two dates in technical specification I'm writing an integration manual that specifies how to use our WCF service for mobile messages and responses. The part in question needs to specify that the search results will be filtered between the start and end dates.  My aim is to make it clear that the time portion of the two dates will be discarded and the result will include everything from the start date to the end date. (In practice the code truncates the dates, adds one day to the end date, and returns the data that conforms to a query something like:  SELECT * FROM Table WHERE DateCreated >= @StartDate AND DateCreated &lt; @EndDate)  But my paragraph looks verbose and clumsy. It doesn't feel right. I feel kind of like John Cleese telling the kids when to put their clothes onto a lower peg. How can I make this less verbose?  Or maybe I'm over-thinking this?     When searching for responses, the search is performed between the supplied dates, after truncating the time portion thereof. The results will include responses with a date-time greater than equal to the start date, and less than one day after the end date. So passing the same value for start and end date will retrieve all responses received on that date, for messages previously sent by that user. Is there consensus in the field of statistics that one book is the absolute best source and completely covering every aspect of GLM - detailing everything from estimation to inference? Chat requirement to post images should be made explicit There is (apparently) a 100 reputation requirement to upload and post images through chat.  There isn't any easily located documentation on this, however. 100 global reputation is listed as the privilege for , while the chat FAQ and help both do not mention any restrictions on uploading images.  As such, could somewhere other than  be used to indicate that there is a reputation requirement in chat for uploading images? My Mac Book Pro 13.-" doesn´t get past the grey booting screen with the apple and the spinning wheel (The loading bar appears but when it´s done nothing happens)   I have 220 GB in one single Partition (Macintosh HD) and there is all my work my music and the only copy of ALL MY PHOTOS. I´m so stupid I don´t have a backup of anything but know I have an external HD with 300GB space. I would like to repair my HDD but more important I need to save my files but I don´t know how to do it in Disk Utility Mode. I read there is a Diskwarrior but I don´t know how to install it without booting the mac. Please help!!!  When I start on Single User mode and enter /sbin/fsck -fs this appears  Executing fsck_hfs (version diskdev_cmds-557-393)  Checking Journaled HFS Plus volume.  The volume name is Macintosh HD  Checking extents overflow file.  Checking catalog file  Incorrect number of thread records  (4, 23745)  Checking multi-linked files.  Checking catalog hierarchy  Invalid volume directory count (It should be 175703 instead of 175701) Checking extended attributes file.  Incorrect number of extended attributes (It should be 340062 instead of 340064)  Checking volume bitmap.  Checking volume information.  The volume Macintosh HD could not be repaired after 3 attempts.  *****The volume was modified *****   What are the next steps if fsck fails to repair a drive? Proving $\arg(zw)=\arg(z)+\arg(w)$ How can I keep multiple live/bootable ISO images on a single USB drive? Suppressing addition of blank page before figure when wrapping long JPEG image I have the following LaTeX file, whose sole purpose is to wrap a JPEG file. This file does not convert well to PDF, so I'm using this unorthodox technique to convert it.  \documentclass{article} \usepackage[pdftex]{graphicx} \pagenumbering{gobble} \begin{document}  \centering\includegraphics[height=650pt]{foo.jpg} \end{document}   This produces a PDF file including the document, but it first includes a blank page, and thus produces a 2 page document, rather than a 1 page document. How do I get rid of this blank page? Using \textheight instead of 650pt as an argument to \includegraphicsmakes the  problem go away, but I'd prefer the larger size I get from using 650pt.  Here is the relevant information from the log file.  &lt;foo.jpg, id=1, 162.6075pt x 1165.35374pt&gt; File: foo.jpg Graphic file (type jpg)  &lt;use foo.jpg&gt; Package pdftex.def Info: foo.jpg used on input line 14. (pdftex.def)             Requested size: 90.6998pt x 650.0pt.  [1  ] Overfull \vbox (100.0pt too high) has occurred while \output is active []   [2 &lt;./foo.jpg&gt;] (./foo.aux) )    I'd prefer to not make the actual document public. Therefore the code above is not a MWE. However, if anyone has tips on how to create a dummy JPEG of the same size, let me know. Can .exe and .msi files (Windows software) be installed in Ubuntu? Is it possible to access a 64-bit dll from a 32-bit application? Why does my query run fast in Environment A, but slow in Environment B? I have a piece of SQL that seems to run really fast in Environment A, but the exact same query runs really slow in Environment B!  Environments are supposed to be the same, so what should I do and/or where should I look in order to see why the query doesn't perform the same? Is there a formula for $(1+i)^n+(1-i)^n$? "I am going to have a walk" My teacher said "I am going to have a walk" today in class and to me it seems incorrect. Correct variants that I know are:    Going for a walk Going to take a walk Going to walk  ...   Is this correct? If so, why? Baffling Elevation Profile out of Profile Tool Plugin When using the Profile Tool plugin I get a graph that looks like the 2nd derivative of what I want. That is, it looks like the rate of change of the slope of the elevation profile. I just want the elevation profile! The data is present for good 1' contour lines. Why is the Profile Tool interpreting the data in such a bizarre way? What am I missing? And what are the Min and Max units?