For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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1answer
183 views

Find bases given that P is the change of coordinates matrix from this to this [Lay P244 Q4.7.19]

Lay P289: Let $V$ be an $n$-dimensional vector space, let $W$ be an $m$-dimensional vector space, and let $T$ be any linear transformation from $V$ to $W$. To associate a matrix with $T$, choose ...
0
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1answer
15 views

proof of uniqueness (hint) $f(m,n)=n+1$ only if $n \ge m$

I'm a bit confused about this proof. let define on $\Bbb N$ a binary fucntion $f$ that satisfies (1) $f(m,n)=n+1$ if $n \ge m$ and (2) $f$ is commutative if I write the values of $f$ in a 2x2 ...
0
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3answers
57 views

Strategy to solve inequalities

I want to prove $|x + y| = |x| + |y|$ iff $xy \ge 0$. I don't understand a place to start. I was thinking of solving using contradiction but, as I am new to real analysis, I don't understand it. ...
1
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1answer
92 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
2
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1answer
103 views

Curl Proof Question

Prove the given formula. So far I have $f\textbf{F}=(f\textbf{F}_1, f\textbf{F}_2, f\textbf{F}_3)$, but I'm not sure where to go from there. Could anyone give me some pointers? Thank you.
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0answers
40 views

Gradient Proof Question

Prove the given formula ($r=||{\textbf{r}}||$ is the length of the position vector field $\textbf{r}(x,y,z)=x\textbf{i}+y\textbf{j}+z\textbf{k}$). $$\nabla \dfrac{1}{r} = \dfrac{-\textbf{r}}{r^3}$$ ...
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2answers
315 views

An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$

Let $c\in\mathbb{R}\setminus\{ 1\}$, $c>0$. Let $U_i = \left\lbrace U_{i, 0}, U_{i, 1}, \dots \right\rbrace$, $U_i\in\mathbb{R}^\mathbb{N}$. We know that ...
0
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1answer
82 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
1
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1answer
57 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\displaystyle \underline{\int_0^1f}$ and $\displaystyle ...
2
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1answer
65 views

The Fundamental Theorem of Calculus and Derivatives

How do I show this in a convincing manner? I know I need to use the Fundamental Theorem of Calculus, but I find it difficult to show any steps in between, as it appears obvious?
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2answers
34 views

Proof relating to the order of $a \mod n$?

The proof required is to show that $\operatorname{ord}_n(a^j) \mid\operatorname{ord}_n(a)$, for any positive integer $j$. I have considered using a proof by contradiction, but am having trouble going ...
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0answers
40 views

A compendium of proof-techniques per objective

Please consider this as an on-going list of techniques preferably per objective or subject. Many mathematical books (at least lately) are focusing on "design patterns" if you like of proof-techniques ...
0
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0answers
88 views

series-parallel system reliability equation proof

I am trying to device a proof for the series-parallel system reliability equation. The mathematical form of the equation is as follows: $Pr(\bigcap_{i=1}^{N}\bigcup_{j=1}^{M} A_{ij}(t)) = ...
1
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1answer
18 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
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1answer
30 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
0
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2answers
33 views

Different way showing a subgroup is a subgroup of another subgroup

http://crazyproject.wordpress.com/2010/04/11/subgroups-and-quotient-groups-of-solvable-groups-are-solvable/ Lemma 1: Let $G$ be a group and let $H,K,N \leq G$ with $N$ normal in $H$. Then $N \cap K$ ...
0
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1answer
24 views

How do I work out the last sentence in this section of a proof of the Unique Factorization Theorem?

The last sentence states that the number of possibilities is $2\log_2 n$ (see the below image to follow the proof). I don't understand how to get $2\log_2 n$ but I understand everything that comes ...
3
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2answers
141 views

Proving Injectivity $x + \sin(x)$

I'm trying to prove injectivity of a particular function (without calculus), but I've come across a bit of a problem. The function is: $$f(x) = x+\sin(x)$$ I started by (abiding by common standards) ...
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2answers
66 views

How to prove something at Normal distribution

$X\sim N(\mu,\sigma^2)$. $A,B$ are constants and $A\ne0$. How to prove that $AX+B\sim N(A\mu+B,A^2\sigma^2)$ ? Thank you!
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5answers
96 views

Is proof by algorithm credible?

I have found this question How to prove that the inverse of a matrix is unique? And while the accepted answer is fine I was wondering if it's possible to proof the uniqueness by algorithm. There is ...
1
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1answer
45 views

Prove $A(A+B)^{-1}B = B(A+B)^{-1}A$ where $(A+B)^{-1}$ and $A$ and $B$ are $n\times n$ matricies.

Hi I have been working on this problem for the longest time. Prove: $ A(A+B)^{-1}B = B(A + B)^{-1}A$ We know that A & B exist in real space, and that they are also N x N matrices. It is also ...
2
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1answer
76 views

Do we sometimes have to go “each way” separately for iff proofs?

So, I often enjoy trying to prove "if and only if" statements by only using if and only if arguments. i.e. RTP: $A \Leftrightarrow D$. Proof: $A \Leftrightarrow B \Leftrightarrow C \Leftrightarrow ...
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3answers
31 views

Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
1
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2answers
75 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
2
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2answers
54 views

How to Complete Sketch of a function of two variables $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ ? [Stewart P930 Question 14.7.4]

For $ f(x, y) = 3x - x^3 - 2y^2 + y^4$ $\implies$ $\partial_x f = 3 - 3x^2, \partial_y f = -4y + 4y^3$. Set both equations to 0 $\implies x = \pm $1 and $y = 0, \pm 1$. $1.$ To determine the ...
1
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1answer
128 views

Sketch Saddle Point of a function of two variables $ f(x, y) = 4 + x^3 + y^3 - 3xy$ [Stewart P930 Question 14.7.3]

As regards $ f(x, y) = 4 + x^3 + y^3 - 3xy$, I computed that (0,0) is a saddle point, and (1,1) is a local minimum. So I'm not asking about this, and am asking only about sketching contours. $1.$ ...
2
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1answer
60 views

Prove that these Sets Containing Infinitely Many Incompressible Strings Exist

We define a set $A$ to be special if: $$\liminf_{n \to \infty} \frac{|A^{\leq n}|}{n} = 0$$ I want to prove that there are special recursive sets that contain infinitely many incompressible strings. ...
1
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1answer
251 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
0
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1answer
29 views

T/F prove for modified Ramsey's theorem

By Ramsey's theorem we know that: $\forall k \in \mathbb N : \exists N \in \mathbb N$ that an arbitrary graph $G$ on a set of vertices $\{1,2,...,N\}$ contains $k$ vertices, which represent either a ...
0
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0answers
64 views

can arbitrary norm in $R^n$ be uniformly continuous?

let $d_1$ be an arbitrary norm on $\mathbb R^n$, and let $d_2$ be a max-norm on $\mathbb R^n$ (max-norm : http://en.wikipedia.org/wiki/Maximum_norm) How can I show that $d_1$ : ($\mathbb R^n$, $d_2$) ...
0
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1answer
45 views

Rank Nullity and Dimension relation

How would one prove the relations: $rank S◦T = rankT-dim(kerS ∩ ImT)$ and $nullity S◦T = nullityT+dim(kerS ∩ ImT)$ I understand that the use of rank nullity theorem is required but am confused by ...
39
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7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most ...
0
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1answer
62 views

double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
2
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1answer
69 views

Help needed - Approximating functions with geometric integration and derivation

I've somehow managed to approximate some functions using cheap tricks as geometrically derivating the function and then geometrically integrating an easier equivalent of the derivative (see here for ...
3
votes
5answers
121 views

Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
0
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1answer
33 views

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $?

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $ ? WTS $x \ge |a| \rightarrow x \ge a \land x \ge -a $     Since $|a| > -a$ then we have $x \ge -a$ ...
3
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2answers
143 views

Proof of $ \phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd $

I'd like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I'm right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ ...
2
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2answers
311 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
0
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1answer
137 views

Approximation of $x!$ - Proof needed

By drawing a graph of the geometric derivative of $x!$, $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}$, i guessed that $e^{\left(\frac{\text{d}ln(x!)}{\text{d}x}\right)}\sim_{+\infty}(x+1/2)$. ...
1
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2answers
27 views

Series, limits and convergence.

Theorem $\,\bf3.3.1.\;$ If the series $$\sum_{n=1}^\infty a_n$$ is convergent then $\lim\limits_{n\to\infty}a_n=0$. Proof. Let $s_n=\sum_{k=1}^n a_k.$ Then by the definition the limit $\lim_n ...
2
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1answer
67 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
2
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2answers
55 views

How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
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2answers
48 views

Easier Proof - Union of finite lin-indep subsets of the eigenspaces = a lin-indep subset. [Lay P285 Thm 5.3.7c]

P267 Lemma. Let $T$ be a linear opera $tor$, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. For eacb $i=1,2,\ \ldots,\ k$, let $v_{i}\in E_{\lambda;}$, the ...
1
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1answer
46 views

Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
0
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0answers
30 views

derivative of the composition of two smooth map is the composition of derivatives

I can't type the LATEX so I uploaded the image. By Googling, I found that is the result of the pushforward, but there is no proof so I can't understand. ...
0
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1answer
57 views

Adaptation of this proof of spectral theorem to the complex case

My question is quite simple, I would like to know why we can't use this proof to the complex case, i.e., the operator $T$ is self adjoint on a complex n-dimensional inner product space $V$. Can we ...
3
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3answers
94 views

Inverse of Dirichlet series equality

I stumbled across a formula in here and tried to prove it for myself: $$\frac{1}{L(s,\chi)}=\sum\limits_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^s}$$ However I got stuck. In my attempt I tried to show ...
1
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2answers
103 views

Can we prove the formula for surface of revolution?

This is math. We like to prove things. However, proofs are rigorous processes (for a good reason) and are more than just "that idea looks like it could make sense". I've seen proofs for many ...
1
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1answer
97 views

Cayley's theorem : precision

I have the form of Cayley's theorem that doesn't say any more than : for every finite group G there exists n such that Sn has a subgroup which is isomorphic to G. Now I'd be interested in knowing ...
1
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1answer
43 views

Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. [Lay P160 Ch 2 Sup Q4]

Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To verify this, compute $ (I \color{orangered}{-A} ...