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2answers
61 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
34 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
413 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?
0
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0answers
47 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
71 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
59 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
24 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
25 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 ...
4
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2answers
88 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
votes
1answer
83 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 ...
0
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0answers
42 views

Solve complex equation using geometric interpretation

We're having some pretty simple exercises to practice for upcoming exams, so I wanted to make them just a little bit harder (read: interesting) for myself by not using typical methods you'd see from ...
0
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0answers
25 views

Product of Consecutive Terms of a Geometric Sequence

Suppose $a_n=aq^n$, where $a>0$ and $q>0$. So $(a_n)_{n=1}^\infty$ is a geometric sequence with positive terms. The product of its consecutive terms, say, $$a_0,a_2,\ldots,a_n$$ equals to ...
1
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1answer
64 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!) ...
5
votes
1answer
614 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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1answer
52 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
169 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
votes
1answer
108 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
600 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 ...