Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

learn more… | top users | synonyms (1)

-1
votes
1answer
280 views

Intuition - Theorem - A group homomorphism preserves normal subgroups - Fraleigh p. 149. Theorem 15.16

p. 128, 129. Theorem 13.12. Let $h$ be a homomorphism of groups $G \to G'$. III. If $S \le G$, then $h[S] \le \color{red}{G'}$. IV. If $S' \le G'$, then $h^{-1}[S'] \le G$. p. 149. ...
1
vote
2answers
69 views

Intuition - Quotient Group of Direct Products - Fraleigh ch. 15

Tried http://www.proofwiki.org/wiki/Quotient_Group_of_Direct_Products Proof on p. 3 and 4 . For the case $n = 2$. Define $h: A_1 \times A_2 \rightarrow \dfrac{A_{1}} {B_{1}} \times \dfrac ...
0
votes
1answer
96 views

Collapsing a Factor to the identity element - Fraleigh p. 14 Theorem 15.8

p. 146: We should acquire an intuitive feeling for this theorem in terms of $\color{red}{collapsing}$ one of the factors to the identity element. p. 147 15.8 Theorem: $\hat{H} = \{(h, e) ...
0
votes
0answers
163 views

Intuition of Picture - Collapse, Factor Group, Homomorphism, Normal Subgroup - Fraleigh p. 144 Figure 15.1

Let $N \unlhd G$. In the factor group $G/N$, the subgroup $N$ acts as identity element. Regard N as being collapsed to a single element, to the identity element. This collapsing of N together ...
2
votes
1answer
239 views

Motivation for Conjugate transpose of a matrix

I'am currently going through a self study of Linear algebra . I'am finding it difficult to grasp the intuition behind the concept of Conjugate transpose of a matrix .Why take the complex conjugate of ...
3
votes
0answers
147 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
1
vote
0answers
231 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like ...
3
votes
1answer
247 views

Equivalences of continuity, sequential convergence iff limit (S.A. pp 106 t4.2.3, 110 t4.3.2)

1. This post became too long, ergo I moved this here. 2. I questioned anew here. How does $\color{red}{(I) \implies (III)}$? This contradicts $a \le b \not \implies \Leftarrow a < b$. 3. ...
1
vote
2answers
84 views

How to motivate vectors as derivations?

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space ...
3
votes
1answer
226 views

Intuition on formal neighborhood in a scheme

Let $X$ be a Noetherian scheme, $x \in X$ a closed point. Denote by $\hat X$ the completion of $X$ along $x$. Now assume that two coherent modules $F, G$ on $X$ coincide over $\hat X$, i.e. $i^*F = ...
6
votes
4answers
537 views

Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$?

I am reading Tom Apostol's Analysis and come across this theorem. Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon >0$? I don't doubt the proof in the book but I don't understand the ...
1
vote
0answers
133 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
0
votes
1answer
508 views

Surface area of a Hypersphere

Hypersphere in 4 dimensions, I am having problem with finding the surface area of it. please help. I know that surface area will have 3 dimensions in 4 dimensional space, I am having trouble to ...
3
votes
3answers
2k views

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
0
votes
1answer
82 views

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwsome number 1 to 17 that I forgotsolns.pdf There are exactly two nontrivial ...
1
vote
1answer
324 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
1
vote
1answer
563 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
3
votes
1answer
99 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
0
votes
1answer
37 views

Can we find an $n$ that minimizes this function?

If we suppose that we have positive integers $k$, $c$, and $v$, can we find the $n$ that minimizes: $$k^n \frac{\log{2^v}}{\log{v}}v^{\log_2{(k \cdot v \cdot c/n)}}$$
131
votes
13answers
21k views

What's the intuition behind Pythagoras' theorem?

Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it. $A^{2} + B^{2} = C^{2}$ $C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ $C^{2} = 25 \text{ cm}^2 + 49 ...
1
vote
2answers
151 views

Intuition — If $k \in \mathbb{Z}$ and $n \ge 2$, then the n$^{th}$ root of k is either an integer or irrational.

Origin — Elementary Number Theory — Jones — p25 — Exercise 2.4 (1) How do you prefigure the answer? Proofwiki start after prefiguring it. (2) What's the intuition? This answer ...
2
votes
1answer
68 views

Intuition for an open mapping

What is an intuitive picture of an open mapping? The definition of an open mapping (a function which maps open sets to open sets) is simple sounding, but it's really not as easy to picture as the ...
9
votes
1answer
479 views

How to change variables in a surface integral without parametrizing

This is a doubt that I carry since my PDE classes. Some background (skippable): In the multivariable calculus course at my university we made all sorts of standard calculations involving surface ...
2
votes
2answers
90 views

How hard is finding values such that

We can work with powers of some naturals $(x_k)^{m_k}$. Here we have $n$ naturals, and $m_k$ is an integer in the range $-r$ to $r$. My question is, how small can $p$ be so that ...
-1
votes
1answer
65 views

If $\phi[H] \subseteq H'$, homomorphism from G to G' induces homomorphism from G/H to G'/H' - Fraleigh p. 143 14.39

Let $H \trianglelefteq \text{ group } G$ and let $H' \trianglelefteq \text{ group } G'$. Let $\phi$ be a homomorphism of G into G'. Show that if $\phi[H] \subseteq H'$, then $\phi$ induces a natural ...
0
votes
2answers
909 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
1
vote
1answer
329 views

Prove we can speak of the smallest normal subgroup containing any subset - Fraleigh p. 143 14.31,32

http://www.auburn.edu/~huanghu/math5310/alg-hw-ans-13 (I think).pdf Apologies if I missed some backslashes which are induced by InftyReader version 2.9.7.2. Does ...
4
votes
1answer
300 views

Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
3
votes
0answers
137 views

Intuition behind a proof showing a square is homeomorphic to a quotient of an interval

There's a rather simple proof for the following theorem: There exists an equivalence relation $\sim$ on the unit interval $I=[0,1]$ such that the quotient $I/{\sim}$ is homeomorphic to the unit ...
2
votes
1answer
200 views

intuitive interpretation of Lie algebra

As you know, the isomorphism between $SO(2)$ and $e^{i\theta}$ allows an intuitive visualization of the Lie algebra $\mathfrak{so}(2)$ as the line $ti$. I wanted to know if there was a similar ...
3
votes
1answer
97 views

Difference between the simplicial nerve and the nerve of a simplicial category

In Jacob Lurie's Higher Topos Theory book, he defines the following notion of a simplicial nerve: Definition 1.1.5.5. Let $\mathcal{C}$ be a simplicial category. The simplicial nerve ...
4
votes
8answers
322 views

Evaluating $\int \frac{1}{\sqrt{x^2 + a^2}}\, dx$ without resorting to trigonometric $u$-substitution

I am looking for a quick and intuitive way to evaluate this indefinite integral without resorting to any trigonometric functions. I'm not sure if it is at all possible to do so, but I was just ...
0
votes
1answer
114 views

understanding into algebraic terms difference between homology and cohomology

my previous question understand quotient group was related to understanding of quotient group,i dont need to know too much detailed in group theore,just some part of algebraic topology,especially ...
2
votes
2answers
127 views

Intuition - Fundamental Homomorphism Theorem - Fraleigh p. 139, 136

Let $\phi: G \to H$ be a group homomorphism with $K = \ker\phi$. Then $G/K \simeq \phi[G]. $ The hinge to the proof is to define $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$. Then we must ...
2
votes
1answer
871 views

Estimate length of confidence interval

From Hogg & Tanis, 8th ed., p. 291: Let $X_1, X_1, \dots X_n$ by a random sample of size $n$ from the normal distribution $N(\mu, \sigma^2)$. Calculate the expected length of a 95% confidence ...
2
votes
4answers
414 views

Is there a more intuitive way to understand matrices and linear algebra?

When learning linear algebar I often blindly follow algorithms to perform operations with matrices without fully understanding concepts behind them. It helped a lot when I started dealing with ...
3
votes
2answers
426 views

Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
2
votes
1answer
131 views

Visualize meaning of quotient in quotient map, group - etc?

What are the reasons for the name "Quotient" in Quotient map, group - etc? Overhead picture shows each of the three cosets in $A_4$ is mapped to a single - gray - node. But this isn't division? ...
3
votes
0answers
220 views

Visualize normal subgroup, normalizer, cosets.

A few important aspects of the relationship $H \lhd N_G(H) \le G$ are highlighted in Figure 7.31. First, the size of $N_G(H)$ is some multiple of |H|, and the size of G is some multiple of $N_G(H)$, ...
4
votes
2answers
206 views

Visualize Fundamental Homomorphism Theorem for $\phi: A_4 \rightarrow C_3$

Question 1. How do you see $\ker\phi = V_4 $ = Klein 4 group ? Book doesn't give formula for $\phi$? Question 2. What's $H$ in $i(aH) = \phi(a)$? I think $H = \ker\phi$ ? Question 3. Why is $i: ...
1
vote
0answers
122 views

About $f(s)=\sum_{a^2+b^2>0} \frac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}=0$ and the Extended Riemann Hypothesis.

Let $s$ be a complex number with a strictly positive real part ($Re(s)>0$). Let $f(s)=\sum_{a^2+b^2>0} \dfrac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}$ where the sum runs over all positive integers $a,b$ ...
1
vote
0answers
65 views

Can we find a reduced row-echelon form for these matrices?

Starting with a Vandermonde matrix: $$V = \begin{bmatrix} 1^1 & 1^2 & 1^3 & \dots & 1^n \\ 2^1 & 2^2 & 2^3 & \dots & 2^n \\ 3^1 & ...
3
votes
0answers
355 views

How Would Arnold Explain the Jordan Normal Form to a 6 Year Old?

How would Vladimir Arnold explain the Jordan normal form, to a six year old, in full detail starting from nothing in a way that somehow explains everything in a deeper way, probably including topology ...
1
vote
1answer
406 views

Unbiased estimator of $\sigma$

This is a problem from Hogg & Tanis, 8th edition, p. 282: Let $X_1, X_2, ... X_n$ be a random sample of size $n$ from a normal distribution. Show that an unbiased estimator of $\sigma$ is $cS$, ...
8
votes
1answer
472 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
0
votes
1answer
174 views

Image of Group Homomorphism is Finite and Divides |Domain of Group| - Fraleigh p. 135 13.44

Let $\phi: G \rightarrow G'$ be a homomorphism. Show that if $|G|$ is finite, then $|\phi[G]|$ is finite and divides $|G|$. Because $φ[G] = \{φ(g) \, | \, g ∈ G\}$, we see $|φ[G]| ≤ \quad |G|$ which ...
0
votes
2answers
62 views

Why are there several roots of complex equations

I'm trying to understand why there are $n$ amount of roots in an equation of the form $z^n=$ complex equations. I understand why there are several answers to a $ \sin(x)=$ equation but I can't wrap ...
1
vote
1answer
283 views

Visualize left, right cosets and conjugation

I drew everything that's in orange. Figure 6.8. Left illustration - Each left coset gH is where H arrows can reach from g, which looks like a copy of H based at g, as in the left illustration. ...
2
votes
1answer
77 views

Visualize cosets of $\left<(0,1)\right>$ partition $C_3 \times C_3$

Page 105 says - A careful look at Figure 6.9 reveals that the cosets of $\left< \, (0,1) \,\right>$ partition $C_3 \times C_3$. How is this true? The picture shows $gH = left picture = ...
0
votes
2answers
117 views

Visualise all vectors perpendicular to one vector and two vectors in R^3 [Strang P19 1.2.6]

I'm only asking about visual/geometric solutions herein. (b) The vectors perpendicular to any vector in $\mathbb{R^3}$ lie on what?. (c) The vectors perpendicular to any two vectors in $\mathbb{R^3}$ ...