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Questions tagged [geometric-interpretation]

Questions about understanding a problem geometrically.

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Corollary of Projection onto a closed convex set and geometric interpretation

I need help with geometric interpretation of this theorem and with the corollary of the theorem: Theorem: projection onto a closed convex set Let $K \subset H$ be a nonempty closet convex set. ...
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simplex $B^{-1} \cdot A_j$ tableau interpretation

In an iteration of the simplex tableau implementation, what is the interpretation of the columns $B^{-1} \cdot A_j$ underneath each variable $x_j$?
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Representation for circumferences intersection

I'm trying to write a formula that represents the intersections points for $n$ circumferences. All of these circumferences intersect them to each other. Is there a good representation to explicit this ...
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3answers
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Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

How can I prove the inequality $$u \leq 2\ln(1+u)$$ for $u \in [0,1]$. What is the geometric intuition behind this inequality?
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2answers
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What's the geometric interpretation of this “vector cross product”?

This answer on StackOverflow answers a question about intersection of two segments. Right at the beginning, it introduces a “vector cross product”. Define the 2-dimensional vector cross product $\...
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1answer
59 views

What is $\zeta(i \infty )$ if $ \zeta(\infty)=1 $ and what is its geometric interpretation?

If we want to compute $ \zeta(\infty) $ for large enough real number we can get $1$ as claimed by Wolfram alpha here which means $ \lim_{s\to \infty} \zeta(s)$ with $s$ is a complex number with nul ...
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What is the geometric meaning of this null-determinant?

While reading about interpolation I came across the following equation in Norlund. It involves determinants and I don't understand it in full yet. I do know how Lagrange and Newton follow by using the ...
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a weird geometrical interpretation comparing harmonic series to inverse square series

I was watching this passage from a course on history of maths, where harmonic series, inverse square series and other ones are described in the context of the more generic zeta function: zeta ...
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2answers
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Interpretation of result of covariance of two functions of two random variables

This question is slightly unusual because I've obtained the answer but I am looking for an interpretation of it as it's not intuitive. If we have $X,Y : Uniform(0,1)$ and independent, let $Z=max(X,Y) ,...
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161 views

Geometrical interpretation for the sum of factorial numbers

I am in need of a way to represent the sum $1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$ in a geometrical way. What I mean by this is that for example, the sum $1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 =...
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Calculate Arc from Center of One AABB to Intersections of Another

I am working in the XY plane with two AABBs: the play space and a game object. My goal is to generate a random direction along their intersection going from the center of the game object in towards ...
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2answers
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Geometrically interpreting $\Im ({z}/{(z + 1)^2} )$

Here is the question from Visual Complex Analysis by Needham. Show geometrically that if |z| = 1 then $\Im\left(\frac{z}{(z + 1)^2}\right) = 0$ What other points apart from the unit ...
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Massey product used to show that Borromean rings are linked

I'm trying to understand an example in "Elements of Homology Theory" from V.V. Prasolov (p. 85-88) where he shows that the Borromean rings represented by three spheres $S_1, S_2, S_3$ in $S^3$ are ...
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2answers
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What's equal $(x+y)^{\sqrt{2}} $ to and what's its geometric interpretation?

I'm confused how I can evaluate $(x+y)^{\sqrt{2}}$ using the Newton binomial. The Newton binomial Newten has positive integer exponents not irrational numbers, so I'm curious to check evaluation of $...
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Geometric interpretation of the tensor product of two projectif spaces $\mathbb{R} P^n \otimes \mathbb{R} P^n$ [duplicate]

We define a Projective space of a vector space as follow : http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces Given a vector space $V$ over a field $\mathbb{K}$, its ...
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3answers
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Geometric or matrix intuition on $A(A + B)^{-1}B = B (A + B)^{-1} A$

I am curious about a seemingly simple identity in matrix algebra. Though matrix multiplication is not commutative (the classic example of noncommutativity, it does allow a commutativity of sorts ...
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Multiplying a matrix with its eigenvectors stretches or contracts the vector without changing its “direction”. Is this true for complex eigenvalues?

I tried to prove this as follows - Suppose A is a square matrix with a complex eigenvalue $\lambda$ and its corresponding eigenvector x. Then, by definition Ax = $\lambda$x Angle between x and the ...
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What is the difference between these two wave equations?

From here, there are (at least) two different wave equations with variable wave speed. Either $c^2(x)$ is outside the Laplacian: $$ \begin{cases}u_{tt} - c^2(x) \Delta u = 0 \quad \textrm{ in } \...
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1answer
77 views

Geometric interpretation of elliptic curve point addition in projective space

I am familiar with the idea of considering an elliptic curve in projective space, particularly thanks to this excellent Crypto SE post. What I find most satisfying about it is the fact that it unifies ...
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1answer
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Intuitive basis for linear transformations (2D/3D matrices)

Can every possible linear transformation, at least in 2 or 3 dimensions, be expressed as a simple sequence of scaling, rotation, stretch-squeeze, and reflection?
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1answer
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Geometric interpretation of Hölder's inequality

Is there a geometric intuition for Hölder's inequality? I am referring to $||fg||_1 \le ||f||_p ||f||_q $, when $\frac{1}{p}+\frac{1}{q}=1$. For $p=q=2$ this is just the Cauchy-Schwarz inequality, ...
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How to understand the exponential operator geometrically?

Consider the geometric interpretation of an orthogonal matrix, a projection matrix, a (Householder) reflector, or even just matrix-vector multiply in general. A matrix takes a vector from a vector ...
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4answers
469 views

Understanding Linear Algebra Geometrically - Reference Request

I know geometry and I know linear algebra but when I understand a linear algebraic concept geometrically, my head just explodes and things just become so much clearer and easier to understand...not to ...
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1answer
61 views

Interpolation using multiple neighboring points

I am wondering, what is the best way to do an interpolation based on $4$ points neighborhood with knowing their value and distance. Here is the illustration: I'd like to know the value of the $x$. I ...
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1answer
24 views

Deriving principal component out of cosine similarity

Cosine similarity is defined as $\frac{A\cdot B}{|A||B|}$ Now, if you multiply it by $|B|$, so that you have $\frac{A \cdot B}{|A|}$, what is the name for this? Could this be considered a ...
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Can you resolve this contradiction concerning the geometric interpretation of differential equations?

In the 1st lecture of the MIT OCW differential equations course, at around the 25th minute, the example y’=-x/y is visualized with multiple isoclines, which intersect at (0,0). That is at (0,0) it ...
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1answer
82 views

Understanding notation for Holder's inequality with the counting measure

I am trying to understand how Holder's inequality applies to the counting measure. The statement of Holder's inequality is: Let $(S,\Sigma,\mu)$ be a measure space, let $p,q \in [1,\infty]$ with $...
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1answer
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What does it mean intuitively for a random variable to be a continuous function from its sample space?

Let's say we have a probability space $(\Omega, \sigma, P)$. and a random variable $Y:\Omega\to \mathbb R$ $P$ is of course a mapping $P:\Omega\to \mathbb R$. Now, Omega is not necessarily a topology,...
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1answer
78 views

Geometric interpretation of fact about vectors

I'm vetting a Linear Algebra book for possible use in a class, and there's an exercise that asks an interesting question. The first task is to show that, for any two vectors $\bf{x}, \bf{y}\in\Bbb{R}^...
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2answers
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Geometric interpretation of $|\frac{z+1}{z-1}| < 1$

What is geometric interpretation of $z \in \mathbb{C}$ such that $\left|\frac{z+1}{z-1}\right| < 1$?
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Is there a physical interpretation of norms greater than L2?

In case anyone is confused about the terminology, here's where I'm pulling it from: http://blog.christianperone.com/2011/10/machine-learning-text-feature-extraction-tf-idf-part-ii/ To summarize the ...
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Physical interpretation of convolution obtained from the solution of a linear first order DE: The bank application example.

Let's image the linear first order differential equation \begin{gather} y'(t) = ay(t) + q(t) \end{gather} that reflects the growth rate of a money bank application, where in the $t=0$ the amount of ...
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615 views

(Geometric intuition) Line integral over vector fields

I'm trying to understand the geometric intuition behind the definition of the line integrals over vector fields. The definition is given below: Definition: Let $\vec{F}$ be a continuous vector ...
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Physical interpretation of gradient descent

Introduction Here are some high-level intuitions that seem to be folklore in the optimization community: The gradient descent method is often motivated from a physical point of view, as a 'ball ...
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Geometric Interpretation of Rearrangement Inequality

We know that many of the famous classical inequalities have geometric interpretations. Can you give a geometric interpretation of Rearrangement Inequality? Note: Rearrangement Inequality is $$...
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1answer
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What is the geometric interpretation of $\displaystyle\lim _{s\to 1}\sin²(\zeta(s))+\cos²(\zeta(s)) $?

it's seems that $\displaystyle\lim _{s\to 1}\sin²(\zeta(s))+\cos²(\zeta(s)) $ dosn't exist as shown here by wolfram alpha , and because $\zeta(1)$ is undefined , My question is to seek if the titled ...
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1answer
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Truly intuitive geometric interpretation for the transpose of a square matrix

I'm looking for an easily understandable interpretation for a transpose of a square matrix A. An intuitive visual demonstration, how $A^{T}$ relates to A. I want to be able to instantly visualize in ...
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1answer
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Geometric interpretation of a basic identity in complex analysis

Consider the identity $|z_1+z_2|^2+|z_1-z_2|^2=2(|z_1|^2+|z_2|^2)$. The proof follows from breaking up the L.H.S. using $|z|^2=z\bar{z}$, expanding the factors and cancelling put the common terms. I'...
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Geometric interpretation and proof of a well-known algebraic statement

We have to show that : Given pairs of integers $\{a,b\}$ and $\{c,d\}$, there exists a pair of integer $\{u,v\}$ such that $(a^2+b^2)(c^2+d^2)=u^2+v^2$. Further, if $a,b,c,d$ are non-zero, and both of ...
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1answer
61 views

Non-geometric interpretation of dot products and eigenvectors

The idea is to motivate the SVD for use in a recommender system. Consider a matrix $A\in \mathbb{R}^{f\times u}$ where $A_{ij}$ captures how user $j$ rates film $i$ (on a scale from 1-10, some ...
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1k views

Hyperplane equation intuition / geometric interpretation

Hello I'm trying to get a geometric appreciation for this n-dimensional hyperplane equation : $\frac{1}{\left\lVert \hat w \right\rVert} \times\ (\hat w \dot\ \hat x) - d = 0$ where: x is a point ...
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2answers
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What is a geometric interpretation of all these information?

We have the tableau $\begin{pmatrix} \left.\begin{matrix} 1 & 0 & \alpha \\ 0 & 1 & \beta \\ 0 & 0 & 0 \end{matrix}\right|\begin{matrix} c\\ d\\ 0 \end{matrix} \end{...
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What is the geometric interpretation of $\mathbb{R}^{\infty}$ or function as :$f:\mathbb{R}\to \mathbb{R}$?

The dimension is the number of coordinates needed to specify a point on the object. For example, a rectangle is two-dimensional, while a cube is three-dimensional. The dimension of an object is ...
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Geometric interpretation of the quotient and remainder when dividing a polynomial by a quadratic

Let $p(x)$ be any polynomial with real coefficients, and $d(x)$ a monic quadratic polynomial. By the division algorithm we may write $$p(x) = d(x) q(x) + r(x)$$ where $\deg r(x) < 2$. Here, $q(x)$...
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Coordinates versus vectors in functions

Since I have had a course in linear algebra I have the following question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^2$ How should I interpret this function? 1) As all ...
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Interpretation of a projection on one eigenspace (from symmetric matrices) to another

I have a reference $N \times N$ symmetric matrix -- with distinct eigenvalues -- decomposed using SVD as: $$ R_{ref} = V_{ref} D_{ref} V^{-1}_{ref} $$ If i get a matrix $S$ -- with distinct ...
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Geometric significance of “circum” property of a spherical triangle

Asked this before. What is the geometrical significance of the relation $$ \frac{ \sin A}{ \sin a}=\frac{ \sin B}{ \sin b}=\frac{ \sin C}{ \sin c}= ? $$ in a spherical triangle? How can we see it ...
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Why does multiplication act like scaling and rotation of a vector in the complex plane? [duplicate]

I regularly use the geometric analogy of multiplication by a complex number to represent a scaling and rotation of a vector in the complex plane. For a very simple example, ...
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picture / geometrical interpretation of mapping cylinder

Can someone give me a geometrical interpretation (picture) of mapping cylinder of a continuous map $g: X \to Y$, where $\operatorname{Cyl}(g) = Z \cup_f Y$, where $Z = X \times [0, 1],\, A = X \times ...
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Gauss-Bonnet theorem proof considering membrane Force and hydrostatic fluid Pressure equilibrium

Is it possible to prove Gauss-Bonnet Theorem by using physics (Mechanics of materials) models? For example in mechanics could one consider static equilibrium by action of hydrostatic pressure ...