Row/Column Space of a matrix
Finding maximal sublist of linearly independent vectors
Implicit newline at the end of each paragraph? "Any" followed by singular or plural countable nouns? This question has troubled me for ages despite my several attempts of looking it up in dictionaries or usage books. Do we say, "Do you have any ideas" or "Do you have any idea"? I do see an example where "any" means "it doesn't matter who/which/what", therefore "You can borrow any book you like." Also, does it matter between using it in questions and negatives? Pluralization rule for "five-year-old children", "20 pound note", "10 mile run" Why are year, pound and mile in the singular form in the phrases below?   five-year-old children  20 pound note 10 mile run   Is that because they're acting as adjectives, which are always invariable in English?  Is it incorrect to say...   five-years-old children? 20 pounds note? 10 miles run? It is well know that $SO(n)$ is connected and $O(n)$ has two connected components: $O^+(n)=\{A\in O(n):\det A=+1\}$ and $O^-(n)=\{A\in O(n):\det A=-1\}$.   In what book can I find this property? How can you encode a string to Base64 in JavaScript? I have a PHP script that can encode a PNG image to a Base64 string.  I'd like to do the same thing using JavaScript. I know how to open files, but I'm not sure how to do the encoding. I'm not used to working with binary data. Help evaluating a gamma function Is there any way of modifying the preamble, so instead of typing   $\mathtt{x}$ ... \begin{eqnarray}     \mathtt{y} &amp; \mathtt{=} &amp; \mathtt{z^\pi} \end{eqnarray}   I can omit the \mathtt{...} commands? Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system.  Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles.  How does String Theory account for it? Heuristically, and mathematically? Is it proper to use "the" before the name of a government organization? I've been looking around on the internet and I can't find anything in depth on how the cable/connection functions. I've only been able to find simple pin out descriptions.  If someone can link me or point me in the right direction, I would be very grateful. Find the limit of $\frac{n^{x}}{(1 + x)^{n}}$ Let $x&gt;0$. Find $$\lim_{n\to\infty}\frac{n^{x}}{(1 + x)^{n}}.$$   What I was thinking involved an inequality, by reason of considering the fact that  $$(1 + x)^{n} &gt; C(n, k) x^{k}$$ for some $k$ such that $n&gt;k$. Expanding the binomial coefficient and taking the $n$ common from each bracket, we have something like $$(1 + x)^{n} &gt; C(n, k) x^{n} &gt; \frac {x^{k}n^{k}}{k! 2^{k}}.$$ But now, I am totally clueless. By intuition, I know that the limit has to be zero, so I just want to prove using the concept of inequalities that what I thought was the right approach (being the sandwich theorem). Any help would be appreciated! I have 24 reputation points, but SO chat is still giving me the message "you must have 20 reputation...." Why can't I join the chat? How do I disable "Link Preview" in Chrome? How to get the valid URL of a Link field from within a Twig template? I have a Link field named field_my_link (machine name).  Within a Twig template I can get the link's URL value with this:  {{ node.field_my_link.uri }}   If inside the Link's URL there is an external URL, e.g. http://example.com it works good.  But, if inside the Link's URL there is stored an internal URL, e.g. "/node/11", I get a value like this:  "internal:/node/11"   with the "internal:" prefix.  How can I get the valid URL?  For example, if the link's URL is "http://example.com" I want "http://example.com" (this already works), if the link's URL is "/node/11" I want the relative URL "/node/11", if the link's URL is "&lt;front&gt;" I want the URL "/", etc.    EDIT  I can get the value I want within a PHP preprocess function, using this:  $url = $variables['node']-&gt;get('field_my_link')-&gt;first()-&gt;getUrl();   How can I access the same value within the Twig template?  I know I can do this within the preprocess function:  $variables['my_url'] = $url;   Then access the url within Twig using {{ my_url }}, but I would avoid to write a preprocess function each time I have a link field. Does it make sense to combine PCA and LDA? Assume I have a dataset for a supervised statistical classification task, e.g., via a Bayes' classifier. This dataset consists of 20 features and I want to boil it down to 2 features via dimensionality reduction techniques such as Principal Component Analysis (PCA) and/or Linear Discriminant Analysis (LDA).  Both techniques are projecting the data onto a smaller feature subspace: with PCA, I would find the directions (components) that maximize the variance in the dataset (without considering the class labels), and with LDA I would have the components that maximize the between-class separation.  Now, I am wondering if, how, and why these techniques can be combined and if it makes sense.  For example:   transforming the dataset via PCA and projecting it onto a new 2D subspace transforming (the already PCA-transformed) dataset via LDA for max. in-class separation   or    skipping the PCA step and using the top 2 components from a LDA.   or any other combination that makes sense. Is there any hint that the MoM or anywhere else got support in combating voldemort the first time around? Or is everything completely UK-centric?  I can't remember anyone in the second war being specifically sent from europe to help, even though we see international wizards in book 4. But I don't think all the events of the first war are covered in the books.  So are other bodies like the MoM involved in the first war, or was it isolated in the UK? When it's OK to omit "where?" Can't update because the word prox* is blocked A weighted version of random.choice What is the importance of eigenvalues/eigenvectors? What is the importance of eigenvalues/eigenvectors?