Prove the orthogonal complement of the row space of $A $is ${0}$ implies $Ax = 0$ has only the trivial solution
I am trying to understand how to find the orthogonal complement of a subspace $M$ of a vector space $V$.   From my understanding, $M^\perp$ is also a subspace of $V$ where all its vectors are perpendicular (orthogonal) to the columns of $M$, which would mean that the dot product of those vectors with each column of $M$ is $0$.  However, how do I find the $M^\perp$ subspace, when $M = \{ \vec{x} \in \mathbb{R}:  2x_2 + 3x_1 = 0\}$? In general, if there's a general approach, how does one attempt to find $M^\perp$ of an arbitrary $M$?  From my understanding, in this case, $M$ is the set of vectors in a plane that satisfy that equation, where $x_1$ and $x_2$ should respectively be the first and second components of the vector $\vec{x}$.
When to use 'the' in front of a country? Sugar as a defence against bacteria? mentions that sugar "at a certain level acts as a preservative". I've always been taught that microorganisms eat sugar and expel acids, that is why sugary food are damaging to teeth. How is it that sugar acts as a preservative, then?  Googling the question I've found   though I don't credit the sources as being especially reputable. What is the real reason that sugar can act as a preservative? Trying to connect to the internet, no WiFi symbol in the top right Why doesn't planet Earth expand if I accelerate upwards when standing on its surface? According to General Relativity I am being accelerated upwards by planet earth while writing this question. But a curious person on the the other side of the planet relative to me would have the same experience. That means we are accelerated in opposite directions, although earths diameter do not seem to increase. How can this be? When do I have to use 'will' instead of 'going to'? Does going to only express an intention and will some kind of prediction that doesn't necessarily happen?  EDIT: Thanks for all your answers. I asked this question because I always fail to complete exercises like this from an ESL book:   We  will fly to Venice in June. (Correct: But, how can I be sure that I'm not sick in June?) Philipp will be 15 next Wednesday (Correct: Since his birthday is a fixed day, this will happen). They will get a new computer. (Wrong: Why?, If I need a new computer, I simply get a new one) In 2020 people are going to buy more hybrid cars.  (Wrong, Why? Who can tell for sure). This material is covered in detail in Dickson's "Linear Groups with an exposition of the Galois Field Theory", chapter XXII and Huppert's "Endliche Gruppen", chapter II, paragraph 8. Since I don't speak german and Dickson's treatment often requires deciphering, I was wondering if there is a "modern" account of this somewhere. rotatebox symbol in caption generates error Following minimal example doesn't work and generates an error  \documentclass{article} \usepackage{graphicx} \usepackage{amssymb} \begin{document} \begin{figure}   FIGURE \caption{$\blacksquare$ result of; \rotatebox[origin=c]{45}{$\blacksquare$}}% \end{figure} \rotatebox[origin=c]{45}{$\blacksquare$} \end{document}   The error message says:  rotatebox-in-caption.tex|7| Argument of \@caption has an extra }. || &lt;inserted text&gt;  ||                 \par  || l.7 ...; \rotatebox[origin=c]{45}{$\blacksquare$}}   Do I have to wrap the \rotatebox in a certain box to make it work? Can you add new statements to Python's syntax? Can you add new statements (like print, raise, with) to Python's syntax?  Say, to allow..  mystatement "Something"   Or,  new_if True:     print "example"   Not so much if you should, but rather if it's possible (short of modifying the python interpreters code) Writing letters of recommendation for more than one student to the same department Suppose multiple students have asked me for letters of recommendation to the same graduate department in the US. Am I expected to explicitly compare the students to one another? Am I supposed to do this in both letters (assuming they haven't been submitted yet)? Do I need to explicitly state a preference for one over the other, or can I just get away with listing relative strengths and weaknesses (for instance, "X is stronger academically, but Y has more research experience and is a stronger programmer")? What is a mixin, and why are they useful? Can you catch a native exception in C# code? In C# code can you catch a native exception thrown from deep in some unmanaged library? If so do you need to do anything differently to catch it or does a standard try...catch get it? Python metaheuristic packages Does QGIS support true circular curves? ASP.NET MVC Custom Error Handling Application_Error Global.asax? How did Newton discover his second law? I've always assumed/been told that Newton's 2nd law is an empirical law — it must be discovered by experiment. If this is the case, what experiments did Newton do to discover this? Is it related to his studies of the motion of the moon and earth? Was he able to analyze this data to see that the masses were inversely related to the acceleration, if we assume that the force the moon on the earth is equal to the force the earth exerts on the moon?   According to , the  reads:     Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.   Translated as:      Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.   My question is how did Newton come to this conclusion? I get that he knew from  the idea of inertia, but this doesn't instantly tell us that the change in momentum must be proportional to the net force. Did Newton just assume this, or was there some experiment he performed to tell him this? Under which assumptions a regression can be interpreted causally? First, don't panic. Yes, there are many similar question on this site. But I believe none gives a conclusive answer to the question below. Please bear with me.  Consider a data generation process $\text{D}_X(x_1, ... , x_n|\theta)$, where $\text{D}_X(\cdot)$ is a joint density function, with $n$ variables and parameter set $\theta$. It is well known that a regression of the form $x_n = f(x_1, ... , x_{n-1}|\theta)$ is estimating a conditional mean of the joint distribution, namely, $\text{E}(x_n|x_1,...,x_{n-1})$. In the specific case of a linear regression, we have something like $$ x_n = \theta_0 + \theta_1 x_1 + ... + \theta_{n-1}x_{n-1} + \epsilon $$ The question is: under which assumptions of the DGP $\text{D}_X(\cdot)$ can we infer the regression (linear or not) represents a causal relationship? It is well known that experimental data does allow for such interpretation. For what I can read elsewhere, it seems the condition required on the DGP is exogeneity: $$ \text{E}(x_1, ... x_{n-1}|\epsilon) = 0$$ The nature of randomisation involved in experimental data ensures the above is the case. The story goes then to argue why observational data normally fails in achieving such condition, reasons including omitted variable bias, reverse causality, self-selection, measurement errors, and so on. I am however uncertain about this condition. It seems too weak to encompass all potential arguments against regression implying causality. Hence my question above. UPDATE: I am not assuming any causal structure within my DGP. I am assuming that the DGP is complete, in the sense that there must be some causality (an ontological position it could well be debated), and all relevant variables are included. The key is to identify the set of assumptions which ensure me causality goes from certain variables to the other, without assuming from the outset such direction of causality.  Many similar posts on the site spend time mentioning why correlation does not imply causation, without providing hard arguments about when it does. That is the case, for instance, of  very popular post. Additionally, in the  on the site about the topic, the accepted answer gives the very vague answer:  Expose all hidden variables and you have causation.  I do not know how to translate that to my question above. Neither is the second most upvoted answer. And so on. That is why I believe this post does not have an answer elsewhere. Solving a system of ODEs with the Runge-Kutta method How do I turn off some formats in GeoServer's WMS? I have published a couple of layers on my Geoserver.  To be complaint with the T&amp;C of my data provider, I need to block the access of some of those layers as Vector data. Hence I have already blocked the WFS services on these Layers.  I came to know that even on the WMS service, you can get Vector Data, if you use GeoRSS and some of the .   How do I block GeoRSS, SVG &amp; UTFGrid as formats in the WMS Service? Simultaneous collaborative editing of a LaTeX file Specify CSS class names on SVG paths using Illustrator Is there a way to edit SVG files in Illustrator where you can specify a CSS class for each path?  I already know in Illustrator that if you give the layer name an actual name, Illustrator will use that name as the path's ID, which is fine if you don't plan on reusing the icon more than once on the same page.  My current workaround is to just use the IDs method, but then later convert the IDs to classes in my code editor, but it's pretty annoying to have to do every time I generate my SVG sprite.  If that's currently not possible in Illustrator, are there any other apps that will allow you to specify that? Or maybe a Grunt or Gulp package?  It looks like you might be able to do it with Inkscape with a , so I might look at that if there's no other good solutions out there.