Hot answers tagged motivation
46
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 ...
21
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 ...
20
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 ...
17
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 ...
14
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 ...
14
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 ...
13
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
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 ...
12
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 ...
10
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 ...
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
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 $\mathbb{R}^n$) ...
8
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 ...
8
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 ...
7
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 ...
7
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 ...
7
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 ...
7
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 ...
7
The quadratic reciprocity law in any of its forms shows that there is an un-obvious correlation between different primes. The $(p,q)$ symbol constrains the $(q,p)$ symbol. This is astonishing compared to other more "linear" theorems about congruences or unique factorization. In its 20th-century reformulations quadratic reciprocity is seen as an avatar of ...
7
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 ...
7
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 ...
6
To "fill the holes" or "add the missing points" would presumably mean embedding the metric space as a subspace of a larger metric space. To avoid trivialities like placing a line inside the plane, it is required (and it appears to be the only sensible interpretation) that the given space is dense in the larger space: every neighborhood of a point in the ...
6
There is a trivial 1-1 correspondence from $\mathbb N=\{1,2,\dots\}$ to $\omega=\{0,1,2,\dots\}$ given by $f(n)=n-1$. This mapping is a bijection, an order isomorphism, an isometry, and preserves limits, in the sense that $f(n)\to\infty$ if $n\to\infty$ (although it would be hard not to satisfy this). Thus practically every property we care about is ...
6
Ultimately, the abstraction in 5 above is well encapsulated in the Yoneda lemma. In a general category, there's no way to dig into the "internal structure" of an object. On the other hand, one can in practice for many concrete categories (sets, groups, topological spaces, ...) and even not-so-concrete ones (simplicial sets, opposite categories of concrete ...
6
Group cohomology was initially observed in nature: people noticed that the homology of certain spaces with trivial higher homotopy groups depends only on the fundamental group, and with considerable ingenuity managed to predict purely algebraically what that homology is from the fundamental group itself.
In a sense, then, people found that $H^1$ can be ...
6
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 ...
6
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 ...
5
With the risk of being redundant, I'll attempt to give my answer, if for no other reason than to help myself. Then again, the best people to answer this question are probably people who use but don't exclusively study mapping class groups.
The mapping class group is a group one associates to a surface, and it's true that it distinguishes two surfaces that ...
5
If a metric $d$ on a vector space $V$ is translation invariant, and compatible with scalar multiplication, in your sense, then define $$\|x\| = d(x,0)$$I claim that this is a norm:
$\|x\|=0$ iff $d(x,0)=0$ iff $x=0$
$\|\lambda x\| = d(\lambda x,0) = d(\lambda x,\lambda 0) = |\lambda| d(x,0) = |\lambda| \|x\|$
$\|x+y\| = d(x+y,0) = d(x,-y) \leq d(x,0) + ...
5
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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