Let $\mathcal{M}_{n,p}(\mathbb{K})$ be the set of matrices $n\times p$ with coefficients in $\mathbb{K}$.  Let $A\in\mathcal{M}_{n,p}(\mathbb{Q})$.  We suppose there exists a non zero solution $X\in\mathcal{M}_{p,1}(\mathbb{R})$ to $AX=0$. ($0$ denotes $[0]_{p,1}$)  Show that there exists a non zero solution $X'\in\mathcal{}_{p,1}(\mathbb{Q})$ to $AX'=0$  PS : This is NOT a duplicate of , which gives a zero solution in my case.    I've tried building up the system of linear equations linked to $AX=0$.  I got the result for simple small square matrices, but I can't manage to generalize it to all the matrices.
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have a solution in $\mathbb{Q}^n$?  Where do I start?    I feel I can use this to :  $$x_i=\frac{1}{a'_{ii}}\left(b'_i - \sum_{j=i+1}^{k} a'_{ij} x_j \right).$$    Here is a thought that just crossed my mind, but I'm not sure if this is legal:  Suppose $Y\in \{\mathbb{R}\setminus \mathbb{Q}\}^n$ and $Y'\in\mathbb{Q}^n$ are solutions to $AX=B$. Then $AY=AY'=B$, or $A(Y-Y')=A(Y+(-Y'))=0$, but  addition between $\{\mathbb{R}\setminus \mathbb{Q}\}^n$ and $\mathbb{Q}^n$ is . . .
I am going through Shankar's Principles of Quantum Mechanics and am having trouble finding the free particle propagator $U(t)$ that satisfies $$\lvert\psi (t)\rangle = U(t)\lvert\psi (0)\rangle$$ due to the degeneracy of the $E$ eigenkets. Shankar says that if the Hamiltonian has degenerate eigenvalues, we change the propagator equation from $$U(t) = \sum_{E} \lvert E\rangle\langle E\rvert e^{-iEt/{\hbar}}\tag{4.3.13}$$ to $$U(t) = \sum_{E} \sum_\alpha \lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar} $$ where $\lvert E, \alpha \rangle $ are the orthonormal eigenkets for the $E$ eigenspace. This makes sense to me, because it seems like whenever we deal with degeneracy to use a projection. He also says for a Hamiltonian with no degeneracy to change the sum into an integral, so I imagine that for a Hamiltonian with a continuous, degenerate spectrum we would have this propagator: $$ U(t) = \sum_\alpha\int_\Re dE\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}$$  I tried applying this to the free particle and had some trouble. In the book, after solving for the eigenkets and eigenvalues, $\lvert E\rangle = \lvert p\rangle$ and $p = \pm\sqrt{2mE}$, he chooses the propagator as a function of the momentum eigenvalues $p$ so that $$U(t) = \int_\Re dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m).\tag{5.1.9}$$ I understand how he got to this, but I'm having some trouble because this doesn't appear to be the same expression that I have above. Splitting the integral up, we have $$U(t) = \int_{-\infty}^0dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ The left integral can changed to an integral over $-p$, and the bounds switch from $0$ to $\infty$ such that: $$U(t) =  \int_{0}^\infty d(-p)\lvert -p\rangle\langle -p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ Now we can substitute and interchange $\lvert p\rangle = \lvert E, + \rangle$,  $\lvert -p\rangle = \lvert E, - \rangle$, $E = p^2/2m$, and $dp = \pm mdE/\sqrt{2mE}$ to get: $$U(t) = \sum_{\alpha = \pm} \int_0^\infty dEm/\sqrt{2mE}\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}. $$ This is the result that we are asked to prove in Exercise 5.1.1. Why is this not the same as the first expression for the propagator? Where did the extra $ m/\sqrt{2mE} $ come from? Am I just wrong in assuming that that is the general expression for a Hamiltonian with continuous and degenerate eigenvalues? Retrieve Force.com registered domain Parametric vs non-parametric machine learning methods I looked-up many references and websites and researched on how to determine if a method is between parametric or non-parametric. I came up with below definitions,     A parametric algorithm has a fixed number of parameters. In contrast, a non-parametric algorithm uses a flexible number of parameters, and the number of parameters often grows as it learns from more data.   From .  Moreover, I found,     A parametric model, we have a finite number of parameters, and in nonparametric models, the number of parameters is (potentially) infinite.    From .  And many other methods of determination, though the problem is none of them can helo ones to determine if a certain hypothetical method is parametric or not. (For instance, why k-means' number of parameters are constant but KNN is variable or basically what do we call a parameter and what we do not?) How can I know if a branch has been already merged into master? I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain.       let $a,b\in R$ s.t. $ab\in P$ ,        $$ab+P=P\implies(a+P)(b+P)=P\implies\overline a \overline b=0$$  and as $R/P$ is an integral domain either $a=0$ or $b=0$ $\implies a+P=P$ or  $b+P=P$.        $\therefore$ either $a\in P$ or $b\in P$.   Hence, $P$ is a prime Ideal.    Am I correct in writing this or am I making some error... Is there a BNF grammar of the TeX language? I'm looking for a BNF grammar of the TeX language, does it exist?  EDIT  For those of us who are not computer scientists, a BNF grammar is one kind of formal description of a CFG: .  For those who don't know their Chomsky, a CFG is a , which means (very roughly) that substitutions can be made independently of their use context. A lady and a monster A famous problem:  A lady is in the center of the circular lake and a monster is on the boundary of the lake. The speed of the monster is $v_m$, and the speed of the swimming lady is $v_l$. The goal of the lady is to come to the ground without meeting the monster, and the goal of the monster is to meet the lady.  Under some conditions on $v_m,v_l$ the lady can always win. What if these conditions are not satisfied?  Edited: the monster cannot swim. If the conditions are not satisfied, then monster can always perform a strategy such that the lady will not escape the lake. On the other hand this strategy is not desirable for both of them because they do not reach their goals. As there was mentioned, this deals with undecidability of the problem. On the other hand, if you imagine yourself to be this lady/monster, you can be interested in the strategy which is not optimal. What is it? If there are such strategies in the game theory?  Edited2: My question is more general in fact. If we have a game with one parameter $v$ when two players $P_1, P_2$ are enemies and if $v&gt;0$ then for any strategy of $P_2$ the player $P_1$ wins. If $v\leq 0$ then for any strategy of $P_2$ there is a strategy of $P_1$ such that $P_2$ does not win and vice versa. I am interested in this case. From the mathematical point of view as I have understood the problem is undecidable since there is no an ultimate strategy neither for $P_1$ nor for $P_2$. But we are solving somehow these problem IRL. Imagine that you are a lady in this game - then you would like to win anyway even while knowing that your strategy can be covered by the strategy of the monster. On the other hand, the monster knows that if he will cover all strategies of the lady she will never reach the shore and he will never catch her. I mean they have to develop some non-optimal strategies. I hope now it's more clear. What does unwrapping a model do and why is it important? Can't connect ad hoc network to Windows 8.1 Pro Does excessive markup affect SEO? I bought a Mac (13", early 2011) almost 5 years ago. When I first got it, it was running under OS X 10.7 and it was pretty fast. I've being upgrading it ever since and it's now running under Mac OS X 10.11.  I guess it didn't totally survive the upgrades because, today, there's a lot of bugs, freezes, lags, "spinning beach ball" appearing,... for example, when I write something on a normal program/editor, the cursor freezes for a few seconds, then catches up to what I'm writting.  As you can guess, that's really annoying! I've been thinkning about upgrading the hardware but I don't know what to do: should I change the RAM to 8 GB - 1 333 MHz (currently my computer has 4 GB - 1 333 MHz)? Or should I upgrade to an SSD (which is pretty expensive and difficult to install)?  Thank you very much. How do I install PlayOnLinux on Ubuntu? I know this is a really basic question, but I've just started with some basic C++ programming after coding a few projects with high-level languages.   Basically I have three questions:   Why use pointers over normal variables? When and where should I use pointers? How do you use pointers with arrays? I have close to two hundred images where the subject looks like he just walked out of a sauna. It was 36 degrees with humidex, but he was not that red.   For comparison, attaching another picture, same camera, same lens, auto white balance, next day.       So, two questions, 1) how did that happen?, and 2) how do I fix it? I can bring down the red, but then I have to manually bring back the lip color in 200 pictures.   And next time when I deal with an older guy at 36 degrees C I will be shooting indoors.   *Edit: The camera is Olympus EPM 2. I am now guessing that he may have been THAT red, but my eyes tricked me. The last picture is from the beginning of the shoot before he got very hot and had two glasses of wine AND the background worked better for his skin and clothes. I do not have colour management setup, but I will get on it. how to not commence a command depending on player health? I was just wondering whether the following statement is correct.  Let R be a ring and M a noetherian R module. Then M is finitely generated. Difference between a Structure and a Union Today, we had a math class, where we had to show, that $a_{100} &gt; 14$ for  $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$  Apart from this task, I asked myself: Is there a closed form for this sequence? Since I didn't find an answer by myself, can somebody tell me, whether such a closed form exists, and if yes what it is? Can I find out which tags I have created? This would be interesting to know.  The tags could be highlighted on the users page for example. Omitting article “the” in front of plural nouns I'm helping someone learn English (second language), and I'm having trouble explaining when to use the and when not to. For example:     Students learn better when teachers are passionate about teaching.   versus     The students learn better when the teachers are passionate about teaching.   There's a subtle difference. The first one is a general statement, whereas the second might imply a study where a group of students learned better than another.  Is the difference simply that, general v. specific?