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0answers
34 views

Open sets in $\mathbb C$ and open sets in $\hat{\mathbb C}$

I usually have a lot of trouble with complex variable when it comes to the geometric representation of $\hat{\mathbb C}$ and what happens in there. I have the next exercise and from quick look at it I ...
2
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1answer
58 views

Is there a nice meaning to the geometric triple product?

Using geometric algebra, I can easily find the geometric tripleproduct of three vectors $a,b,c \in \mathbb{R}^3$ to be $$abc = a \left(b \cdot c \right) - b \left( c \cdot a \right) + c \left( a ...
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0answers
35 views

Geometric Interpretation of Fractional Derivatives

I was looking for a geometrical interpretations of fractional derivatives and fractional integrals. I would be glad to see any kind of intuitive and preferably visual interpretation of the objects ...
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0answers
8 views

Interception Solid

Could someone draw for me (in $\Bbb R^3$, $x,y,z$ axes) the resultant solid of the interception of $x^2+y^2 \le 1$ and $y^2+z^2 \le 1$? I'm getting a little tired of thinking about the solid but can't ...
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0answers
22 views

How do I put this into a linear equation?

I really don't know what I would do first. I moved $x$ to the other side of the equal sign to make it $2y = 12 + x$ and then I was going to divide it by $2,$ but i realized you can't divide $x$ by ...
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1answer
48 views

Geometric interpretation of 2D-Translation's Matrix representation

I just learned the trick of writing a translation of a 2-dimensional real vector as a matrix multiplication in a 3-dimensional space - wikipedia explains it here. Basically it shows: $$ ...
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1answer
49 views

Geometric interpretation of determinant of a system of homogenous linear equations

What is the geometric interpretation of the determinant of a matrix representing a system of homogenous linear equations? We know that iff the determinant is equal to zero the system has a ...
0
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2answers
69 views

Are there any formulas or identities that involves $\pi$ but have no obvious trigonometric interpretation?

Every formula that involves $\pi$ has an underlying trigonometric interpretation but it is not usually obvious. I wonder if there are any formula like the Gaussian integral ...
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2answers
28 views

Find the geometric interpretation of general solution

Here are the linear equation: $$2x+z=0$$ $$-x+3y+z=0$$ $$-x+y+z=0$$ I have found that the general solution is, $$t \begin{bmatrix} \frac{-1}2\\ \frac{-1}2 \\ 1 \\ ...
1
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2answers
90 views

physical interpretation or practical meaning of a distance

How can I find/understand the physical explanation or practical meaning of a distance? For example Euclidean distance is the shortest path between two points, how about for other distances?
0
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0answers
106 views

Geometric interpretation of expected value of a random variable

What's geometric interpretation of the expected value of a random variable? What I have understood so far can be elaborated as this: An expected value can be interpreted as the 'center of mass' of the ...
12
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10answers
434 views

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?
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0answers
50 views

Interpretation of the complement $S_5 \setminus D_6'$.

On account of a homework question $$ \text{does there exist an injective homomorphism } D_6 \to S_5 \text{ ?}$$ which I solved positively, I raised the question $$ \text{what does the complement ...
0
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2answers
85 views

Rotation of Matrices and their interpretation

Given are now two matrices and I have to discuss what the given functions are doing (geometrically). Maybe you can revise/add the following: given are the matrices $ A = \begin{pmatrix} \cos(a) ...
0
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2answers
74 views

Interpretation of vector line integrals

I recently came across two types of line integrals which I had not previously come across before and am unsure as to what their interpretation would be: Integrating a scalar field with vector line ...
1
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1answer
26 views

Showing for $f,g:\mathbb{R}\rightarrow\mathbb{R}$ that for $M>0, |f(x)|\le M|g(x)|$ for $x>x_0$

Showing for $f,g:\mathbb{R}\rightarrow\mathbb{R}$ that for $M>0, |f(x)|\le M|g(x)|$ for $x>x_0$. This is a repost of a question that is probably too long to ever get an answer (I feel compelled ...
1
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0answers
30 views

Odly phrased big-oh question, have I done what is required? (because it's not really big-oh, it's more graph sketching)

I've encountered these before, but never phrased or defined as follows, I'd like to know if I've done whatever the question wants to draw attention to (if it didn't want to draw attention to ...
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3answers
168 views

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 ...
5
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1answer
87 views

Can something like $\text{Hom}(V,K)$ be visualised?

I have no trouble visualising vector spaces like $\Bbb R^3$ and (e.g.) a subspace of dimension $2$, which would just be a plane through the origin of a $3$-D space, but I'm having trouble visualising ...
1
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1answer
75 views

Interpretation of Linear Algebra and superposition

I have been told and have seen that knowing Linear Algebra is foundational in pursuing advanced mathematics. After dealing with it enough I have gotten "used to it" (I still need a lot more practice!) ...
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1answer
891 views

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 ...
1
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2answers
69 views

expected value of random variables

Take two random variables $X=a+bX_0$ and $Y=c+dY_0$, and define $T=X-Y=\mu+\sigma Z$ where $\mu$ is the mean of $T$, $\sigma$ its standard deviation and $Z$ is a standardized random variable with mean ...
2
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2answers
176 views

$i^{th}$ root(s) of unity

If we define $S:=\{z\in\mathbb{C}:z^n=1\}$ (i.e. the $n^{th}$ roots of unity), then $|S|=n$ (i.e. we have $n$ of them). We can even go as far as to say: $$S=\{z_k:k\in\mathbb{N}\cap[1,n]\}=\{e^{2\pi ...
2
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1answer
124 views

Sorgenfrey line topology

For every $x\in\mathbb R$ we have $\mathcal{B}(x):=\{[x,z): z>x\}$ as the sorgenfrey line. First I want to show that there is topology $\tau$ defined on $\mathbb R$, i.e I have to show that ...
7
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5answers
840 views

Geometric Interpretation for Definite Integrals with $\pi$ in the result [duplicate]

What is the geometric interpretation for the following integral? What is a nice geometric interpretation for the following integral (possibly in relationship to a circle) that emphasizes why we get ...