Questions about understanding a problem geometrically.

# Questions tagged [geometric-interpretation]

158 questions

**194**

votes

**4**answers

### What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.
For example,…

Zed

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**96**

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### Geometric interpretation of $\det(A^T) = \det(A)$

$$\det(A^T) = \det(A)$$
Using the geometric definition of the determinant as the area spanned by the columns, could someone give a geometric interpretation of the property?

dfg

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**74**

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### Geometric interpretation for complex eigenvectors of a 2×2 rotation matrix

The rotation matrix
$$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$
has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\pmatrix{1 \\ -i}$. The real eigenvector of a 3d rotation…

J..

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**24**

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**1**answer

### 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, for which I have geometric intution: The projection…

Asaf Shachar

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### 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 mention easier to remember or figure out its…

Fixed Point

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**18**

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### Do the BAC-CAB identity for triple vector product have some intepretation?

As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from Vector Analysis - J.W. Gibbs) that's not at all…

pppqqq

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### Geometric Interpretation of the Basel Problem?

Does the Pi in the solution to the Basel problem have any geometric significance? Every time I see Pi, I have to think of a circle. I would love to see a nice intuitive picture connecting the Basel problem with geometrical figures. Anyone here…

theboombody

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**13**

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**1**answer

### 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 space (after a basis has been fixed) and performs a…

Fixed Point

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**12**

votes

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### Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?

user142299

**12**

votes

**3**answers

### Geometry of the dual numbers

A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to the complex numbers $a+bi$, where…

David Zhang

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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 my mind what I'm doing to the space when…

evolution

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**10**

votes

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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
$$x_ny_1 + \dots + x_1y_n \leq x_{\sigma(1)}y_1+ \dots +…

scarface

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**10**

votes

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### Why does multiplication act like scaling and rotation of a vector in the complex plane?

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, i would point up along the Y axis and multiplying it by i again would be a…

William Grobman

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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 normal to a patch, tension in the curved patch…

Narasimham

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**9**

votes

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### 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
$1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
can be represented geometrically as a pyramid…

Sigfrid Stjärnholm

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