# Tagged Questions

For questions about the motivation behind mathematical concepts and results. These are often "why" questions.

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### Simple examples for motivation of topology [on hold]

It is easy to see motivation for groups and fields, as abstractions of operations defined on integers, rationals, reals etc. and how the results from those abstractions apply to integers, reals etc. ...
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### Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
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### Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
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### Motivation behind the definition of tangent vectors

I've been reading the book Gauge, Fields, Knots and Gravity by Baez. A tangent vector at $p \in M$ is defined as function $V$ from $C^{\infty}(M)$ to $\mathbb R$ satisfying the following properties: ...
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### Motivation for construction of cross-product (Quaternions?)

I'm trying to present a narrative that brings the (3D) Cross Product into existence. "Given two vectors $\mathbf u$, $\mathbf v$, how to construct a vector perpendicular to both?" ... looks like ...
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### Motivation for the study of algebraic structures

I am currently studying group theory and I realized that most concepts we study are just definitions on which we build theory. I do understand that some theorems are beautiful and don't need any ...
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### Why is Grobner basis useful? [duplicate]

I don't get the motivation for calculating Grobner bases. What's good by computing a Grobner basis for an ideal of $k[X_1,...,X_n]$? Moreover, is there any theorem whose proof relies on the use of ...
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### Motivation for/history of Jacobi's triple product identity

I'm taking a short number theory course this summer. The first topic we covered was Jacobi's triple product identity. I still have no sense of why this is important, how it arises, how it might have ...
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### Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendieck then later gave a more abstract definition of the right derived ...
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### Motivation for introducing $A^{\prime} := \frac{1}{\left|G \right|} \sum_{g \in G} \Delta(g^{-1}) A \Delta(g)$ in representation theory proof

Assume $\left|G \right|= \left| G \right| \cdot 1_F$ is invertible in $F$. Let $\Delta:F \to GL_n(F)$ be a representation and $U \subseteq F^n$ be an $F$-subspace that is $\Delta$-invariant. Then ...
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### Hopfian and Co-Hopfian Modules. [closed]

Let $M$ be a $R-$module. We say that: $M$ is a Hopfian module, if every epimorphism of $M$ is a monomorphism. $M$ is a Co-Hopfian module, if every monomorphism of $M$ is an epimorphism. Why do we ...
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### Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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### Why were limits introduced in calculus? [closed]

I have got the intuitive of the limits but why were they introduced in calculus, are they really helpful to us?
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### What is the 'meaning' of nowhere dense set?

In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres). So what is the 'meaning' (i.e motivation, ...
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### What are the Big Theorems in Information Geometry?

I am working on preparing a talk on information geometry to a young finance/applied math audience. Motivating this area is turning out to be a little difficult. What are some big theorems or results ...
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### Providing motivation for the importance of the concept of a 'basis'.

In a few situations, I found myself being asked by younger students why the concepts of a basis was important. First, in the concept of linear spaces, it's easy to explain that having a basis allows ...
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### Purpose of a vector space [closed]

I am currently studying linear algebra and I've seen what vector spaces are, but I can't seem to understand what their purpose is. What do they allow us to do?