Questions tagged [geometric-interpretation]

Questions about understanding a problem geometrically.

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Geometric effects of parameters in homogeneous second-order ordinary differential equations with constant coefficients

In homogeneous second-order ordinary differential equations (ODE) with constant coefficients, of the form: \begin{equation} ay''+by'+cy=0, \end{equation} is there any conclusion about the effects of ...
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Ricci curvature in 2 dimensions

A well known formula for interpreting Ricci curvature is the following (see for example Wikipedia's article). Consider the Taylor expansion of the volume form at a point $p$ in normal coordinates $\...
theflame's user avatar
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Knowing inverse matrix based upon a picture?

I know how to calculate an inverse matrix (by creating the augmented matrix and Gaussian elimination to get the identity matrix) and I know how to do it for a general $2 \times 2$ matrix by taking two ...
Ally's user avatar
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Derivative of velocity with respect to position

Preface: I am a physicist, not a mathematician, but I'm seeking something closer to mathematical rigor. I'm deriving the expression for kinetic energy from Newton's second law, and it seems to rely on ...
concertpi's user avatar
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How to avoid Cusps and Corners when generating spline?

I already asked this in robotics.stackexchange.com, but no reply. So I've decided to ask here Here I wrote Qubic spline trajectory generation. The following are the result. Spline that generated ...
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Geometric interpretation of vector and transpose?

$v^Tv$ is the dot product of a vector with itself, which is just its norm squared, an intuitive geometric quantity. Is there something to be said about $vv^T$? Is there some kind of relationship ...
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Geometric interpretation of an underdetermined system of equations

I am having a difficult time connecting different parts of a geometric interpretation of an underdetermined system of equations. Given the matrix $$A = \begin{bmatrix} 2 & 1 & 3\\ 1 & 2 &...
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Geometric interpretation of the Hessian

Assume we have a smooth function $f:\mathbb{R}^2 \to \mathbb{R}$. We may then form the differential of $f$, denoted by $Df$, given by the row vector $$ Df=\Big[\frac{\partial f}{\partial x_1},\frac{\...
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Geometric interpretation of inner product

I already know the geometric interpretation of dot product (the length of projected vector multiply with the other vector's length) But is there a geometric interpretation for any inner product ?
Faulheit's user avatar
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What is convolution, how does it relate to inner product?

Let's consider the formula of the inner product between two continuous functions: $$<f,g> = \int_{-\infty}^{+\infty} f(t) \cdot g(t) \ dt$$ The inner product is intuitively defined to give some ...
Ramzi Baaguigui's user avatar
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Geometrical interpretation of the complex map $ f(z)=\frac{1}{(z-i)^n} $

In a complex variable class, a professor asked us the following. Describe the function $$ f(z)=\frac{1}{(z-i)^n} $$ geometrically. I tried writing $z=x+iy$ to see what the function does to lines and ...
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The spectral norm of the difference of two rank-$1$ matrices is equal to the sine of their angle [closed]

Can someone help me understand either geometric intuition or mathematical proof that how for two unit vectors $a$ and $b$, the following holds: $$\left\lVert aa^T - bb^T \right\rVert_2 = \sin \theta$$ ...
Zulqarnain's user avatar
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How to geometrically interpret the gradient of this linear equation?

I can't actually picture in my head how the gradient w.r.t $\theta$ in this equation works geometrically. For example, for $x, y \in \mathbb{R}^d$ and $\theta \in \mathbb{R}^{d \times d}$ $$\begin{...
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Geometric Interpretation of the Exterior Derivative of a 1-form

I was reading this link where says that the geometric interpretation of the exterior derivative of a 1-form $\varphi$ is “the sum of $\varphi$ on the boundary of the surface defined by its arguments” ...
user1234567890's user avatar
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Interpretation of $l_p$ norm inequality

If $1\le p\le q\le \infty$, we know that the following inequality holds: $$\|a\|_q\le \|a\|_p.$$ What could be a possible interpretation of this inequality for a non-mathematician? For example, can we ...
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Any geometric interpretation for the adjoint system of a linear dynamical system?

On page 26, Section 1.3, of his book on linear dynamical systems1, Professor Roger Brockett asks: If $$\dot{\mathbf{x}}(t) = A(t) x(t) , \qquad \mathbf{x}(0) = \mathbf{x}_0$$ and $$\dot{\mathbf{p}}(t)...
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Geometric interpretation of $A^TA$

For a transformation $A \in \mathbb{R}^{n\times m}$ what exactly is the geometric interpretation of the transformation $A^TA$. If I understand it correctly the entries of $A^TA$ are the inner products ...
jonithani123's user avatar
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Geometrical meaning of $x^2+y^2+z^2-xy-xz-yz$

I am looking for a geometrical interpretation of the symmetrical expression $$f=x^2+y^2+z^2-xy-xz-yz\tag{1}$$ with $x,y,z \in \mathbb{R}$. I could for example $f$ interprete as dot products of a ...
granular_bastard's user avatar
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Is there a geometric interpretation of $\int_a^b\left(x-\frac{a+b}{2}\right)f'(x)dx$

For a differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$, using integration by parts, we get $$\frac{(b-a)(f(a)+f(b))}{2}-\int_a^bf(x)dx=\int_a^b\left(x-\frac{a+b}{2}\right)f'(x)dx$$ The LHS ...
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What does "interpret the point" mean?

As far as I understand, interpreting a function means finding its vertex, determining its shape and the direction, in which the function's hands are pointing. For example, if I am asked to interpret ...
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Understanding the Graph of a Multinomial Distribution

I am trying to understand exactly what information the graph of a multinomial distribution is supposed to convey. The thing I find strange is that a binomial distribution is graphed in two dimensions ...
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How to interpret strongly/weakly infinite dimensional spaces and what are some examples?

I have a question regarding strongly infinite dimensional spaces. Loosely speaking, $X$ is called strongly infinite dimensional if any pair of closed disjoint sets can be separated by a subset $L_i$ ...
Tereza Tizkova's user avatar
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Geometrical interpretation of variance in terms of moments

Suppose $X$ and $Y$ are discrete random variables such that $$Y = aX+b$$ where $a$ and $b$ are scalars. It is easy to find a geometric interpretation of the following two facts $$\Bbb E[Y] = a \,\Bbb ...
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What is the geometric interpretation of $ \frac{1}{2}||a||^2\leq \langle a,b\rangle $?

Give $a, b$ in $\mathbb{R}^n$. What is the geometric interpretation of the following? $$ \frac{1}{2}\|a\|^2 \leq \langle a,b\rangle $$ In other words, what criteria should $a$ and $b$ have to satisfy ...
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What is the geometric interpretation of matrix addition?

I was studying linear algebra and trying to get a visual "feel" for it through watching 3Blue1Brown's "Essence Of Linear Algebra" series here Essence Of Linear Algebra Here, matrix ...
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Geometric interpretation of row operations on matrix (when solving systems of equations)

I understand that 2D matrix representing the lines in 2D space gives a unique solution where those lines intersect. Same in 3d, unique solution is where the planes intersect. Can someone explain what ...
Retko's user avatar
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Generalizing the geometric interpretation of dot product to simple $k$-vectors

Background: For $u, v \in \mathbb R^n$, the dot product $u \cdot v$ can be interpreted geometrically as follows: Its magnitude is the product of the lengths of $u$ and $\operatorname{proj}_{u} v$. ...
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Philosophy of Integration (geometric interpretation): the difference between dx & a point?

This may sound like a stupid question, but if you're familiar with "Infinite Hotel Paradox", probably it won't be; So here we go: Integration of a scalar function $f: \mathbb{R} \rightarrow \...
Captain Husayn Penguin's user avatar
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The radius of the circle which just touches the outer circle

Suppose two concentric circles with radius $a$ and $b\ (>a)$ and origin as their center. I wanted to put another circle whose center lies in a line $x=a$ (that is red line) in such a way that it ...
Young Kindaichi's user avatar
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2 answers
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Where does the 3rd power of the denominator of the curvature of the graph of a function come frome?

According to Wikipedia, the curvature of the graph of a function $f$ is given by the following ratio (assuming second-differentiability). $$\mathscr{k}_f(x) = \frac{f^{\prime\prime}(x)}{(1 + [f^{\...
Galen's user avatar
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Remove outliers from a noisy curve

Allow me to present some images so that I may explain my problem. The images on the left contain a smooth curve surrounded by noise. Is there any approach that would help in eliminating the noise and ...
Sau001's user avatar
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1 answer
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Geometric interpretation of differentiability

I know that the geometric interpretation of differentiability for a function $f:\mathbb{R}^2\to \mathbb{R}$ in a point $(x_0,y_0)$ is that it admits a tangent plane in the point $P=(x_0,y_0,f(x_0,y_0))...
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Why doesn't a line passing through origin satisfy the 2 intercept form of the equation of a straight line?

The 2 intercept form of an equation of a straight line, is: x/a + y/b = 1 Where, a and b are the x and y intercepts respectively. For a line passing through origin, these two intercepts will become 0, ...
ihateelectricalphysics's user avatar
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1 answer
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Geometric interpretation of the non-uniqueness of the solution of a system of linear equations

Show that the equation $$\pmatrix{3&-7&0\\2&2&5\\1&3&4}\pmatrix{x\\y\\z} = \pmatrix{3\\2\\1}$$ does not have a unique solution, and give a geometrical interpretation. Now, the ...
yumemc2's user avatar
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3 answers
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What is the Point, and Meaning, of the Mean Value of a Function?

Ok, so I know that the mean value of a function, $f(x)$, on the interval $[a,b]$ is given by (or defined by?) $$\frac{1}{b-a}\int_a^bf(x)~dx$$ but I have $2$ basic questions about this: $1$: From a ...
A-Level Student's user avatar
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Where is the "Interior" of a Clifford Torus

[I do not believe I have tagged this entirely correctly, feel free to change them.] In the case of spheres there is a related structure called a ball which based on its definition I visualise as being ...
Disgusting's user avatar
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Is there a geometric interpretation for $\det(\nabla X)$?

Let $X$ be a vector field on a Riemannain manifold $M$. Consider $\nabla X:TM \to TM$, where $\nabla$ is the Levi-Civita connection of $M$. We know that $\operatorname{tr}(\nabla X)=\operatorname{div} ...
Asaf Shachar's user avatar
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Dirac Delta Function of a complex variable

Is there a context where delta(x+i*c), where c is a real number, makes sense? It came up while I was doing an Inverse Fourier Transform, and I failed to appreciate its significance. Does anyone know ...
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Geometric interpretation of linear programs with both inequality and equality constraints

I recently do a homework with linear programming, the standard form as: $$\begin{array}{ll} \text{minimize} &\displaystyle \mathbf{c}^T\mathbf{x}+\mathbf{d}\\ \text{subject to} & \mathbf{A}\...
jason's user avatar
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Geometric solution of $au_x+bu_y+cu=0$

This is a question that I read, and I am curious about how to solve $au_x+bu_y+cu=0$ by interpreting geometrically. Additionally, the second answer on that page says Let $v(x,y)=e^{cx}u(x,y)$ and ...
stoneaa's user avatar
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Trying to Understand Time Interpolate

I am trying to understand pandas interpolation, time interpolate. I know that it is the same as linear interpolation if the time indexes are equally spaced. But I don't understand how it works when ...
dddd_y's user avatar
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3 votes
1 answer
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Geometrical interpretation of matrix $A-B$

Is there a geometrical interpretation of subtraction of two matrices, with a special case of $I -A$ (subtraction of a matrix from identity matrix)? Reference: If $A$ is an idempotent matrix, the range ...
Monalisha Bhowmik's user avatar
2 votes
0 answers
57 views

Is there a geometric interpretation of eigenvalues of integer matrices?

In some instances, like physics, you may find that quantities you are after are eigenvalues of matrices. However, for example, explaining that "the mass of a muon is an eigenvalue of a matrix&...
zooby's user avatar
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Geometric intuition of the solution to $A \bf{x} = \bf{b}$

I am seeking the geometric intuition behind the following two cases using matrix $A_{m \times n}$ with rank $r$. When $m>n>r$, $A\textbf{x} = \textbf{b}$ has a solution if $\textbf{b}$ is in $C(...
Weizi Li's user avatar
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Geometric intepretation of transpose of matrix [duplicate]

Whenever we see a matrix $A=\bigl( \begin{smallmatrix} 3 & 2 \\ 1 & 2 \end{smallmatrix} \bigr)$ and $v=(3, 2),$ we can visualize that $(3, 2)$ represent the coordinates of $\mathbf i$ vector ...
Newton Nadar's user avatar
1 vote
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basic level maths to geometrical interpolations (trilinear, prism, pyramid, tetrahedral) - where do I start?

I'm attempting to read Computational Color Technology by Henry R. Kang but I only have a GCSE level understanding of maths (if even that, it was 15 years ago..). I'd like to learn about geometrical ...
Harry Bennett-Snewin's user avatar
1 vote
1 answer
318 views

A question about derivatives between Euclidean spaces: how should we construct it and interpret its definition?

As it is known from the single-variable calculus, given $X\subseteq\textbf{R}$, a function $f:X\to\textbf{R}$ and a adherent point $x_{0}\in X$ which is also a limit point, we define the derivative of ...
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How can 2 independent variables lead to a single variable parametrization? (Intuition)

Take the following formula: $$ \mu_{ij}=\left(\alpha_i\Sigma_i^{-1}+\alpha_j\Sigma_j^{-1}\right)^{-1} \left(\alpha_i\Sigma_i^{-1}\mu_i+\alpha_j\Sigma_j^{-1}\mu_j\right). $$ Where the alphas are ...
Makogan's user avatar
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What it is the interpretation of the below picture in set theory?

somone sent me the below picture and He asked me to give him its interpretation regarding set theory ? but I ask alos about its geometricall interpretation ?
zeraoulia rafik's user avatar
2 votes
1 answer
161 views

Examples about Cohen-Macaulay property of rings and book recommendation on intuition

My professor asks me to give an example about a local non-CM rings that are CM after modding out at any minimal prime. After a while failing to do so, I believe it is impossible since there exists a ...
Neil Vaen's user avatar