I'm not asking how matrix multiplication is done. I know how to multiply matrices and can look at two matrices and say whether or not they can be multiplied. I just want to know what matrix multiplication reveals to us.
Multiplication of matrices &mdash; taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product &mdash; is not a very intuitive operation: if you were to ask someone how to mutliply two matrices, he probably would not think of that method. Of course, it turns out to be very useful: matrix multiplication is precisely the operation that represents composition of transformations. But it's not intuitive. So my question is where it came from. Who thought of multiplying matrices in that way, and why? (Was it perhaps multiplication of a matrix and a vector first? If so, who thought of multiplying them in that way, and why?) My question is intact no matter whether matrix multiplication was done this way only after it was used as representation of composition of transformations, or whether, on the contrary, matrix multiplication came first. (Again, I'm not asking about the utility of multiplying matrices as we do: this is clear to me. I'm asking a question about history.)
What does the word treats mean in this context? I don't think it means entertainment, food, drink etc.. Maybe it is an old usage of this word? Does it mean influencing factors or something like that?  This is the first part of the title of the first chapter of the novel "Oliver Twist" by Charles Dickens. The whole title reads:     Treats of the Place Where Oliver Twist was Born and of the Circumstances Attending His Birth How can I upload files asynchronously with jQuery? I have an unlocked North American (Canada) iPhone 5s, where can I buy a data only SIM card in Japan ? I'm not interested in renting nor in a voice plan, I'll be staying for 4 weeks.  I'll be landing in Tokyo, so preferably around there. Can I get them from the airport somewhere ? I'm assuming I won't have problems with the network/frequencies, am I correct ? Torque on a rod? I've learned that the torque on an object is just $ \text{perpendicular force}\times\text{radius}$ and that has worked well for things like seesaws but it doesn't take into account the torque of the object itself!    In the picture the rod has a total mass of 5kg, and by definition the torque on the right is greater than that on the left.  $$T=Fr$$   I am supposed to solve the question     What distance to the left of the fulcrum (triangle supporting the rod) would I have to put a 3kg weight to balance the rod?     I've tried calculating the net torque $T_{net} = I\alpha$ but I don't really know how to approach this. I'm looking for an explanation more than an answer. Is editing sources.list a good idea? This doubt is hitting me since 2-3 days, so I decided to test it myself. What I did is I installed Ubuntu 16.04.4 in VMWare and edited /etc/apt/sources.list and replaced xenial with bionic using:  sudo sed -i 's/xenial/bionic/g' /etc/apt/sources.list   Then I ran  sudo apt-get update &amp;&amp; sudo apt-get upgrade    I was able to upgrade to 18.04, confirmed by using lsb_release -a. However, I encountered some problems like:   apt does not have a stable CLI interface. Use with caution in scripts. unmet dependency of  python3-aptdaemon.pkcompat Release 'bionic-backports' for 'appstream' not found Broken Unity   I solved all the errors and installed gnome and gdm3.   So, I just want to know is it a good idea to edit sources.list? I know that my system might broke but is there any other specific reason not to use this.  Note: I tested this on Ubuntu 16.04 in VMWare to clear my doubts only. Asymptotic expansion for Fresnel Integrals Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares? Order of nested random effects in lme4 I just fit a model in lme4, and I'm wondering what the heck I fit...  I have individuals id, and each is measured pass/fail on items that can be described using two factors, f1 and f2. My theory says that id and f2 can interact. So I want to compare these models:  # f2 doesn't matter m1 &lt;- lmer(pass ~ f1 + (1|id), family="binomial")  # f2 is a fixed effect, doesn't interact with individuals m2 &lt;- lmer(pass ~ f1 + f2 + (1|id), family="binomial")  # f2 is a nested random effect, interacting with individuals m3 &lt;- lmer(pass ~ f1 + (1|f2/id), family="binomial")   First, am I right that there's a random effect of each level of f2, and each individual's level of f2 is partially pooled, depending on the number of observations for that individual?  Second, does it make sense to fit f2 as a fixed effect with random individuals nested within each level of f2? How does one write that?  Third, what's this, with f2 and id reversed in the formula? I typed it by accident, but now I want to understand it.  m3weird &lt;- lmer(pass ~ f1 + (1|id/f2), family="binomial") My grandparents bought a flight on United Airlines that departs on Feb 10 2019, as the one and same ticket, from Toronto, to ORD with 1 55-minute layover, then Maui. Please ping me if I ought post their return flight on Feb 24's details.   advises:     There should be no customs and immigrations formalities at ORD, as YYZ is a , and CBP matters will be handled in Toronto.    I'm assuming that  persists during their vacation to Maui.   reported some TSA agents' refusal to work.  Will  refuse to work, or any other federal workers pertinent to travelling to and from the US? Where are the developer options in Android 4.2? Is there any way to view which applications are accessing the Internet on Mac OS X? I think I remember a command line utility that did this, but I can't remember what it's called and a few Google searches turn up nothing. How to solve the equation $\phi(n) = k$? Trigger a button click with JavaScript on the Enter key in a text box Why can't enum's constructor access static fields and methods? This is perfectly valid with a class, but is not allowed with an enum.  What I'm trying to do is store my enum instances in a static Map. Consider this example code which allows lookup by abbreivation:  public enum Day {     Sunday("Sun"), Monday("Mon"), Tuesday("Tue"), Wednesday("Wed"), Thursday("Thu"), Friday("Fri"), Saturday("Sat");      private final String abbreviation;      private static final Map&lt;String, Day&gt; ABBREV_MAP = new HashMap&lt;String, Day&gt;();      private Day(String abbreviation) {         this.abbreviation = abbreviation;         ABBREV_MAP.put(abbreviation, this);  // Not valid     }      public String getAbbreviation() {         return abbreviation;     }      public static Day getByAbbreviation(String abbreviation) {         return ABBREV_MAP.get(abbreviation);     } }   This will not work as enum doesn't allow static references in its constructor. It however works just find if implemented as a class:  public static final Day SUNDAY = new Day("Sunday", "Sun"); private Day(String name, String abbreviation) {     this.name = name;     this.abbreviation = abbreviation;     ABBREV_MAP.put(abbreviation, this);  // Valid } I want to sort in-text citations by year, but bibliography by name. It sound exactly like .  However, suggested solution (using \usepackage[sorting=ynt]{biblatex} and \printbibliography[sorting=nty]) does not work for me (see MWEB below). Indeed, the list of references remains year-sorted instead of name sorted.  Question: How to force sorting=nty in \printbibliography?      \documentclass{article}     \usepackage[%         backend=biber,%         style=authoryear-comp,%         natbib=true,%         sorting=ynt%     ]{biblatex}     \addbibresource{\jobname.bib}     \usepackage{filecontents}  \begin{filecontents}{\jobname.bib}     @book{key2000,       author = {Author, A.},       year = {2000},       title = {Alphabetical fist \&amp; Year last},       publisher = {Publisher},     }     @book{key1900,         author = {Boathor, B.},         year = {1900},         title = {Alphabetical last \&amp; Year first},         publisher = {Publisher}, } \end{filecontents}  \begin{document}     Citation should be year-sorted, even if \verb+\citep+ is random \citep{key2000, key1900}.% citation NOT year-sorted      \printbibliography[sorting=nyt] \end{document}   P.S.: At the opposite, using  \begingroup     \newrefcontext[sorting=nyt]     \printbibliography \endgroup    overrides the \usepackage[sorting=ynt]{biblatex} option (i.e. LoR is name-sorted, but in-line citations as well). Using Regex to generate Strings rather than match them Since the Denoise feature was added to Blender I only use the default setting with it and it really does the work. But sometimes the denoise is noticable on some of my renders. How can I overcome the blurred effect? Denoising for me is a big help because my system is quite aged. Energy conservation in General Relativity How to apply standardization/normalization to train- and testset if prediction is the goal? How do I find out what fonts are used in a document/picture? Is it possible to identify the font used in a specific document/picture?  Answers to this question should identify:   Possible methods to do this (perhaps one answer per method) and adequately describe how to use it (as opposed to merely stating it); Ways of finding the identified fonts, if possible (free or not); and Any prerequisites associated with the method used, if required (for example, "In order to use method X, your document has to be in format Y").   This question is meant as an FAQ, based on an . Its aim is to facilitate the community with the general procedures involved in font identification. Similar cases are solved on a per-usage basis on 's  tag.