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.

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

Let $\phi: G \to G'$ be a group homomorphism with $K = \ker\phi$. Then $\Phi: G/K \to \phi[G]$ given by $\Phi(gK) = \phi(g)$ is an isomorphism. If $\nu : G \to G/K$ is the homomorphism given by ...
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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 ...
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Intuition - Normal Subgroup Test - 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 ...
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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? ...
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Visualize cosets of kernel of homomorphism, normal subgroup

Question 1. 'Since we know that the codomain is a group, this cannot happen.' I don't understand. Can someone elaborate? I know all homomorphisms are functions but not vice versa. Functions are ...
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Visualize every quotient map follows a pattern, subgroup and its left cosets

page 167. Because of the Fundamental Homomorphism Theorem, Nathan Carter calls non-embedding homomorphisms quotient maps. This is one of the key facts about homomorphisms: they come in ...
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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 $N_G(H)$, ...
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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: ...
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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 $a,b$ ...
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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 & ...
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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 ...
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91 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 $cS$, ...
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“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 ...
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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 ...
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Can we convert a sum-of-products into strictly sums?

So I start with a sum-of-products: $$f = \sum_{k=a}^b{\prod_{j=c}^d{g(j,k)}}$$ I'm wondering if we can somehow convert this into a sum of sums, ie: $$f = \sum_{k=a}^b{\sum_{j=c}^d{h(j,k)}}$$ It's ...
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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 ...
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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. ...
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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 = ...
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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 $\mathbb{R^3}$ ...
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How do I convince someone that $\mathbb{R}^2$ and its copy inside $\mathbb{R}^3$ are different?

One of my friends is taking a first course in linear algebra now, and one of the problems on his latest homework was to explain why $\mathbb{R}^2$ and $\{(a_1,a_2,a_3) \in \mathbb{R}^3 \mid a_3 = 0\}$ ...
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Is there a nontrivial homomorphism for each of the given groups? - Fraleigh p. 134 13.38, 13.41, 13.43

(38.) $\mathbb{Z} \rightarrow S_3$? Let $φ(n) = \begin{cases} \mathrm{id} \in S_3 &, \text{for all $n$ even,} \\ \mathrm{transposition} (1,2) &, \text{for all $n$ odd integers.} ...
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Intuition - Homomorphic Image of Group Element is Coset - Fraleigh p. 135 13.52, p.130 Theorem 13.15

Theorem 13.15: Let $\phi: G \rightarrow G'$ be a group homomorphism, $g \in G$. Then $g\ker\phi = (\ker\phi)g = \operatorname{Im}^{-1} \left[ \; \{ \; \phi(g) \; \} \; \right] = \phi^{-1}[ \; \{ ...
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What are the coefficients of these trigonometric sums?

I have two functions that I'm working on. The first is: $$ \begin{align} \cos x &= (\cos 1)^3 \cos(3-x) \\ &{}+ 3 (\cos 1)^2 (\sin 1) \sin(3-x) \\ &{}- 3 (\cos 1) (\sin 1)^2 \cos(3-x) \\ ...
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curvature of helix

Here is the curve of a helix parametrized by its arc length $\alpha(s) = ( a\cos(\frac{s}{c}), a\sin(\frac{s}{c}), b(\frac{s}{c}) ), s \in \mathbb{R}$ such that a$^2$ + b$^2$ = c$^2$. The curvature ...
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Intuition and Strategy - Index of Subgroup of Subgroup Proof - Fraleigh p. 103 10.38

This isn't a duplicate. I tried kb's answer and Answerer 1 but I'm still confounded. I like $\frac {\left| G\right| } {\left| H\right| }$ better than $[G:H]$ hence I write it as a fraction. Suppose ...
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A subgroup has the same number of left cosets as right cosets - Trick - Fraleigh p. 103 10.32, 35

(32.) Let H be a subgroup of a group G and let $a, b \in G.$ Prove or disprove. If ${aH= bH},$ then $Ha^{-1} = Hb^{-1}.$ $\color{blue}{Ha^{−1}} = \{\color{magenta}ha^{−1} | h ∈ H\} = ...
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Magical Counterexamples to Nonidentities - Power of Cosets and Right Coset - Fraleigh p. 103 10.30, 33

Let H be a subgroup of a group G and let $a, b \in G.$ Prove the statement or give a counterexample. If $aH= bH,$ (30.) then $Ha= Hb.$ (33.) then $a^2 H = b^2 H.$ I understand p. 3: Let $G = S_3$ ...
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Cross Product Component Values

When taking the cross product, the x component of the perpendicular vector is the (signed) area of the yz projection of the parallelogram spanned by the two vectors it's orthogonal to-right? And ...
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When can I successfully move differentiation from inside a summation to outside?

So, essentially, I want to start with a summation: $$s = \sum_{k=a}^b{ f(k,x) }$$ and differentiate with respect to $x$, inside the summation. My question is: When does the following equality ...
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Why is the derivative a limit?

Start by assuming that function curves are made of an infinite amount of lines (i.e. look at the image above but instead of approximating it using a finite number of lines, use infinite lines). This ...
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Countably or Uncountably Many Discontinuities

I want to know why the following function has uncountably many discontinuities: $$f(x)=\left\{\begin{array} & x^2 & x \not \in \mathbb{Q} \\ 0 & \text{otherwise} \end{array}\right .$$ ...
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How can you describe topology to a non-mathematician without using continuous deformations?

One of the most frequently used ways to describe topology to non-mathematicians is that it studies the properties of objects that are preserved under deformations where ripping or tearing is not ...
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What is the intuition behind $\Delta_1^0$ sets and $\Delta_1^1$ sets?

In the context of first-order arithmetic, if $\phi$ is a formula with only bounded quantifiers, then if you put existential quantifiers in front it becomes a $\Sigma_1^0$ formula according to the ...
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Colored Picture for Equivalence Classes, Relations, Partitions, .. [closed]

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
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Except the Dividend Itself — Any Divisor is Less than Half of the Dividend

Postulate $d \neq n$ is a divisor, $n$ is a dividend. Why $d \le n/2$? I know the dividend itself is a divisor. $d|n$ is defined as $\exists \; c\in \mathbb{Z}$ such that $dc = n$. ...
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For $n \ge 3$, the only central element of $S_n$ is the identity - Fraleigh p. 86 8.47

Strengthening Exercise 46, show that the only element of $S_{\large{n \ge 3}}$ satisfying $\sigma\gamma = \gamma\sigma$ for all $\gamma \in S_n$ is the identity permutation. I call this $i$. The ...
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Is this statement incorect?

Taken from the introductory chapter of "How to Prove it A Structured Approach": $(2^b - 1)(1 + 2^b + 2^{2b} + ... + 2^{(a - 1)b}$ $=2^b(1 + 2^b + 2^{2b} + ... + 2^{(a - 1)b}) - (1 + 2^b + 2^{2b} + ...
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Why are derivatives lines?

If you look at a function "infinitely close", the difference between two points is a line: __ __/ __/ Where each "__" is a point, and "/" is the ...
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If two powers of permutations are equal and have no common symbols, they're the identity. - Mulholland p. 44 Proof to Theorem 4.2

Theorem 4.2 (Order of a Permutation): The order of a permutation written in disjoint cycle form is the least common multiple of the lengths of the cycles. Proof: One cycle: As we noted above, a ...
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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 ...
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What is known about doubly exponential series?

I've been exploring functions that have a general form: $$\sum_{k=0}^\infty{ a^{b^k} } \tag{1}$$ In particular, I'm now checking this equality, which seems to hold: $$2 \sum_{k=0}^\infty{ \left( ...
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When do Entries Remain after and despite Matrix Multiplication? [Strang P92 2.5.41]

Suppose $E_1, E_2, E_3$ are 4 by 4 identity matrices, except $E_1$ has $a, b, c$ in column 1 and $E_2$ has $d, e$ in column $2$ and $E_3$ has $f$ in column 3 (below the $1$ s). Multiply $L = ...
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All Subspaces of $\mathbb{R^4}$ and $\mathbb{C^n}$ [Strang P129 3.1.14]

The subspaces of $\mathbb{R^n}$ are $\mathbb{R^4}$ itself, three-dimensional planes $\mathbf{n \cdot v = 0}$, two-dimensional subspaces $\mathbf{n_1 \cdot v = 0}$ and $\mathbf{n_2 \cdot v = ...
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Necessary and Sufficient Conditions about the Fundamental Four Subspaces [Strang P143 3.2.34]

I'm trying to ascertain the necessary and sufficient conditions on $A$ and $B$, given that $\mathbf{x \in \mathbb{C^n}}$, for : $1. \, null(A) = null(B), \; 2. \, colspace(A) = colspace(B), \; 3. \, ...
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Relationships between Reduced Row Echelon Form and the Fundamental Four Subspaces [inspired by Strang P143 3.2.34]

I'm trying to apprehend all the links between two matrices' RREFs and their $4$ fundamental subspaces. Does $RREF(A) = RREF(B) $ $1.1.$ $\implies null(A) = null(B)$? True because $null(A) = ...
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Intuitive way to arrive at the maximizing argument for the binomial probability

The binomial probability term $q^{n}(1-q)^{N-n}$ is maximized when $q=n/N$. This can be easily arrived at by differentiating the given probability term with respect to q. Is there a more intuitive way ...
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Intuition for differentiating beneath the integral

I apologize in advance for a vague question. There is a theorem: If both $f(x,s)$ and $\partial _sf(x,s)$ are continuous in $x$ and $s$, then $$\partial_s\int_a^bf(x,s)\,dx=\int_a^b ...
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How can we get a formula for nested multiplications

I have the result of a function, which can probably best be described as the infinite nested function itself: ...
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Intuition - If $Ax = b$ has infinitely many solutions, why can't $Ax = c$ have only one solution? [Strang P165 3.4. 22]

If $\mathbf{Ax = b}$ has infinitely many solutions, why is it impossible for $\mathbf{Ax = c}$ (where $\mathbf{c}$ is a new right side) to have only one solution? Proof : Take two solutions of ...