What is the geometrical Interpretation of a determinant? I know a bit about the geometrical Interpretation of a matrix      but don't have any idea about this
What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know something more, please.  Thank you very much.
Writing a Virtual Printer in .NET I can't find a definitive answer for this. As far as I know, you can't have multiple __init__ functions in a Python class. So how do I solve this problem? Suppose I have a class called Cheese with the number_of_holes property. How can I have two ways of creating cheese objects...  One that takes a number of holes like this: parmesan = Cheese(num_holes = 15) And one that takes no arguments and just randomizes the number_of_holes property: gouda = Cheese()  I can think of only one way to do this, but this seems clunky: class Cheese():     def __init__(self, num_holes = 0):         if (num_holes == 0):             # Randomize number_of_holes         else:             number_of_holes = num_holes  What do you say? Is there another way? Localisation is isomorphic to a quotient of polynomial ring I am having trouble with the following problem.      Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.    I just want a hint.  Basically, I've been looking for a surjective homomorphism from $R[x]$ to $R_D$, but everything I've tried has failed. I think the fact that $f(a)$ is a unit, where $f$ is our mapping, is relevant, but I'm not sure. Thanks Unable to load DLL (Module could not be found HRESULT: 0x8007007E) Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$ I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above? what is the difference between "yet" and "still"?   When we can use "still"? and when we can use "yet"?   Are they synonyms or not?  "The plan could yet succeed." Can I use "still" in that sentence instead of "yet"?  "I still don't understand." Can I use "yet" in that sentence too? Given two sets$$A=\{1,3,5\}\quad,\quad B=\{1,3,8\}.$$ Then I compute the $A\setminus B=\{5\}.$   But my book$^\dagger$said: ... . Also, observe that $x\notin A\setminus B$ does not mean that $x\notin A\lor x\in B$. Why?  I don't know how to explain the Why, please give me some example.    $^\dagger$, second edition, page $4$.  EDIT: My intuition tells me that it does mean that. I guess there is some special case about null-set or about the restriction of the universal set. I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two?  Linearly interpolating between the two almost works, if they are similarly distributed and sized.   provides a forumla at the bottom for combination but it doesn't seem right; two identically distributed distributions should have the same covariance, but the formula at the bottom of the page doubles the covariance.  Is there a way to combine two matrices? A sequence $\{x_j\}$ is said to be fast-Cauchy if $\sum_1^\infty d(x_j,x_{j+1})&lt;+\infty$. Show that every Cauchy sequence has a fast-Cauchy subsequence.   **My attempt:**Argue by contradiction, suppose there is no such subsequence. Then for any subsequence $\{{x_j}_k\}$,  $\sum_1^\infty d({x_j}_k,{x_j}_{k+1})$ diverges. Vaguely recall from an analysis course, if a series diverges, it doesn't not satisfy the Cauchy criterion. That is to say, there exists $\epsilon&gt;0$ such that for all $k$ and all $p\ge1$, $d({x_j}_{k+1},{x_j}_{k+2})+...+d({x_j}_{k+p}+{x_j}_{k+p+1})\ge\epsilon$.  Then I intuitively think that this is a contradiction, since if the original series is Cauchy, then after a large $N$, since $d(x_m,x_n)$ has to be very small, so that their sum should be smaller than given $\epsilon$, which is a contradiction. But I don't know how I can write down the concrete proof? Seeking open source solution to access rasters in File Geodatabase? I'm looking for a way to access raster layers in an Esri File-GeoDatabase without using ArcObjects or ArcPy, specifically I would like to access the raster layers with GDAL. I understand that the GDAL FileGDB and OpenFileGDB give access to vector datasets within File-GeoDatabases, and that ESRI's own FileGDB C++ API does not support raster access.   Is anyone out there aware of any open source solutions? In a 2-columned document, how do you get a table to stretch across the entire width? I'm working on a document that runs two columns of text down the page, like many scientific conference proceedings.  However, I'd like for a table to run across the entire width of the page, as it is far too wide to fit into one column.  How do I do this? is it true if $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant? I am not able to  find out what property of holomorphic map I need to apply. please help.Thank you.  $f(z)=u(x,y)+iv(x,y)$, $\bar{f}(z)=u(x,y)-iv(x,y)$ Tractatus 3.333 and Russell's paradox Can anyone explain to a non-logician how Tractatus 3.333 refutes (or fails to refute) Russell's Paradox? Please explain his use of symbols! Does timbre consist in pitch and volume? I read that the physical properties of a sound wave correspond to its audible qualities: pitch, volume, and timbre. However, an oscilloscope uses only two-dimensions to accurately depict the physical properties of a wave. Intuitively, pitch and volume seem more basic than timbre does, so I surmise that timbre must consist in those two properties.    Does timbre consist in those two properties? How do two-dimensional waveform diagrams depict three properties of sound waves? Using the uncertainty principle to estimate the ground state energy of hydrogen I have been reading through this  of the ground state energy of hydrogen  and others like it. In this one it says it is using the uncertainty principal but then proceeded to use the following: $$pr=\hbar$$ But why is it not using: $$pr=\frac{\hbar}{2}$$ which is more in line with the uncertainty principle and what comes earlier in the derivation. I have a friend who has got a computer that is not connected to the Internet. Is there any way to install software offline easily? Duplicate questions without answers Extrude Faces along Local Normals I am trying to extrude faces of a cylinder along local normals. 3dsmax can do it easy, couldn't find how to do it in blender.  The extrude region option extrudes it as shown in the first example. I want it done like the 2nd example. How come SG-1 couldn't get back to Earth? "Z-level" in TikZ