# Tag Info

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Linear algebra is vital in multiple areas of science in general. Because linear equations are so easy to solve, practically every area of modern science contains models where equations are approximated by linear equations (using Taylor expansion arguments) and solving for the system helps the theory develop. Beginning to make a list wouldn't even be relevant ...

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Having studied Engineering, I can tell you that Linear Algebra is fundamental and an extremely powerful tool in every single discipline of Engineering. If you are reading this and considering learning linear algebra then I will first issue you with a warning: Linear algebra is mighty stuff. You should be both manically excited and scared by the awesome ...

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As far as I know, historically smooth manifolds were the first manifolds studied by people like Poincare, Riemann, up to Whitney. There were a few major events that caused people to take things like topological and PL manifolds seriously, but originally people were not motivated to study these kinds of objects. Here are some of the big events/ideas that ...

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I will try to give a rather elementary motivation which I believe to be close to how this identity was actually discovered. Let us consider the sine function $f(z)=\sin z$. As a function of complex argument, it has one period: $f(z+2\pi)=f(z)$. Further, it is holomorphic in the whole complex plane with only simple zeroes given by $\pi\mathbb{Z}$. It ...

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Topology can mean different things in mathematics, depending on the context. If a mathematician describes themselves as a topologist, this likely means that they study various kinds of shapes (technically, manifolds, or related kinds of spaces), with an eye perhaps to classifying them (a typical example being the classification of closed orientable ...

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The full appreciation of Jacobi's triple product identity can not be done without some understanding of the elliptic functions. However, it is possible to develop some parts of the theory of elliptic functions without any complex analysis. Anyways back to the triple product identity, it says that $\displaystyle \sum_{n = -\infty}^{\infty}z^{n}q^{n^{2}} = ... 15 The "product" of vector spaces is more properly thought of as a direct sum, since the dimension increases linearly according to addition,$\dim(V\oplus W)=\dim(V)+\dim(W)$. Tensor products are much bigger in size than sums, since we have$\dim(V\otimes W)=\dim(V)\times\dim(W)$. In fact, in analogy to elementary arithmetic, we have distributivity$(A\oplus ...

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I don't think there is just one motivation. I will mention three. First, the Jacobi identity says precisely that the bracket is a derivation with respect to itself, where a derivation of an algebra is a map $d$ with $d(a\cdot b)=d(a)\cdot b+a \cdot d(b)$. Thus, writing $\mathrm{ad}(a)$ for the map $b \mapsto [a,b]$, the Jacobi identity may be rewritten ...

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You asked about a natural problem that leads to this integral. Here's a summary of the argument I give in my undergraduate probability theory class. (It's due to Dan Teague; he has the article here.) Imagine throwing a dart at the origin in the plane. You're aiming at the origin, but there is some variability in your throws. The following assumptions ...

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This is a good question. My own take on motivating quadratic reciprocity is recorded here (these are lecture notes from an undergraduate course on introductory number theory). If you look there, you will find that most of what I have said is an elaboration of the two points you bring up. I think a crisp way of explaining what QR does for you is in the ...

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Quadratic reciprocity allows you to make precise certain intuitions about the primes. More precisely, it tells you that for every finite set $p_1, p_2, ... p_n$ of primes and every function $f : \{ 1, 2, ... n \} \to \{ -1, 1 \}$ there exists an arithmetic progression such that any prime $q$ in that progression satisfies $\left( \frac{p_i}{q} \right) = ... 12 TL;DR version Linear algebra is your ticket to multidimensional space. Study it if you are into economics, computer graphics, physics, chemistry, statistics or anything quantitative (in today's world, that's everything). Meaning of "Linear" and why it is "Easy" Since you are asking the question, perhaps you would benefit from a discussion of what ... 12 Motivation can be broken up into essentially two parts: (i) Stable homotopy groups of things appear in nature, and (ii) the stable category is technically convenient. I'll try to give some indication of these below, but none of the bullet points do the topic justice... There's lots of love to be had here. Stable Homotopy Groups Appear in Nature Cobordism. ... 11 I can't speak much to what traditional Spectral Graph Theory is about, but my personal research has included the study of what I call "Spectral Realizations" of graphs. A spectral realization is a special geometric realization (vertices are not-necessarily-distinct points, edges are not-necessarily-non-degenerate line segments, in some$\mathbb{R}^n$) ... 11 There is indeed some nice intuition behind these definitions, and the good news is that not even all that deep. Remember two things: First, that this cohomology all comes by the "fixed by" functor$M\to M^G$, and second, that these crossed homomorphisms come from the definition of cochains, and more directly, the coboundary operator from$n$-chains to ... 10 This question already has a number of nice answers; I want to emphasize the breadth of this topic. Graphs can be represented by matrices - adjacency matrices and various flavours of Laplacian matrices. This almost immediately raises the question as to what are the connections between the spectra of these matrices and the properties of the graphs. Let's call ... 9 The group axioms are intended to abstract the properties of discrete symmetries (that is, bijections from a set to itself). That is, we may define a "concrete group" to be a group of permutations of some set with composition as the group operation. An abstract group is supposed to be a version of a concrete group that does not rely on a choice of group ... 9 One nice example is solving quadratic equations. If you know only about natural numbers or integers then it is difficult to give a concise universal solution for quadratic equations. But if you abstract your notion of "number" to allow negative, rational (possibly even imaginary) numbers, then one can give a single universal formula - the well known ... 8 Topological manifolds arise naturally as the background of other phenomena. For example, nature throws at you examples of spaces which admit several smooth structures and when you try to describe that phenomenon you need to say something like «there are many different$Y$s one can put on an$X$». In the situation of exotic smooth structures, a natural class ... 8 The topological category is inherently beautiful. Some very pretty and subtle phenomena happen. My favorite example of this is that some knots bound locally flat (that is they admit a locally trivial normal bundle) topological disks into the 4-ball, but they don't bound smooth disks. That is, there are topologically slice knots which are not smoothly slice. ... 8 Flatness in commutative algebra satisfies a geometric condition: the fibers of a morphism between two varieties (schemes) don't vary too wildly. A flat morphism$f \colon X \to Y$of varieties (schemes) can be thought as a continuous family of varieties (schemes)$\{ f^{-1} (y) \}_{y \in Y}$. An important theorem says that if$f \colon X \to Y$is a flat ... 8 Spectral graph theory is a discrete analogue of spectral geometry, with the Laplacian on a graph being a discrete analogue of the Laplace-Beltrami operator on a Riemannian manifold. The Laplacian$\Delta$can be used to write down three important differential equations, both on a graph and a Riemannian manifold: The heat equation$\frac{\partial ...

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Tensor products turn multilinear algebra into linear algebra. That's the point (or at least one point). They let you treat different kinds of base extension (e.g., viewing a real matrix as a complex matrix, making a polynomial in ${\mathbf Z}[X]$ into a polynomial in $({\mathbf Z}/m{\mathbf Z})[X]$, turning a representation of a subgroup $H$ into a ...

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The mapping class group (in genus $g$) is the fundamental group of the moduli space of compact Riemann surfaces of genus $g$. Indeed, the latter space is the quotient of a contractible space (Teichmuller space) by the mapping class group. Therefore a lot of the geometry of this moduli space is encoded in the mapping class group. This is explained in the ...

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"Shake it up" a bit! Supplement your text with another text that approaches the topic differently. Also, make some changes in your approach to math: see how what you're learning connects to what you already know, and how each topic you encounter connects to the earlier topics. Try to anticipate what you'll soon encounter by making conjectures using what you ...

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This math.SE question may be relevant, but not pedagogically optimal. Pedagogically I think the simplest answer is to axiomatize topological spaces via the Kuratowski closure axioms. Instead of specifying what properties open sets or closed sets satisfy, the Kuratowski closure axioms specify a closure operator $S \mapsto \text{cl}(S)$ on subsets $S$ of a ...

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Quadratic reciprocity is important because it provides a bridge between two apparently distinct branches of mathematics, namely the theory of Galois representations and the theory of automorphic forms. $L$-functions provide the bridge across the two theories. Let $K=\mathbf Q(\sqrt{D})$ be a quadratic field with fundamental discriminant $D$. Let ...

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Basically almost every phenomenon that is continuously time-dependant and determinist (nothing random) is modelled by a differential equation. There are examples in physics, chemistry, biology, economics, you name it. A few examples among trillions : mecanics : the movement of a point subject to a force F is determined by differential equation \$m \ddot x = ...

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Here's a simple question: why does the product of two groups have the same underlying set as the Cartesian product of sets, but the coproduct of two groups is not even close to the same underlying set as the disjoint union of sets? Well, the latter would be silly, but if you know enough category theory to know that the product and coproduct are dual to each ...

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I'm not going to add nothing directly related to your question and previous answers, but make some propaganda of a theorem I like since I was student and which, I believe, says something stronger than comparing some intuitive notion of completness with its definition. A somewhat related notion of completeness is the geodesical one. The definition may not be ...

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