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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2answers
21 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
0
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4answers
24 views

Proving a recursive sequence is bounded

I'm proving that the limit of the following recursive sequence is $\dfrac{10}{9}$: $$s_0=1,\,s_n=s_{n-1}+\frac{1}{10^n}\quad\text{for }n\ge1$$ Showing that the sequence is monotonic was easy enough, ...
2
votes
4answers
59 views

Proving that $6^{2n+1} + 1$ is divisible by $7$ for $n\geq 1$ by induction

How should I go about solving a problem like this using induction? Would I: First test $(n = 1)$ so that $6^{2(1)+1} + 1 = 6^3 + 1 = 217/7 = 31$. Then assume $(n = k)$ so that you have $6^{2(k) + 1} ...
2
votes
1answer
46 views

Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$

I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$. Also, how is this different from $f_n \to f$ uniformly on $\left(0, ...
0
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1answer
31 views

Proving relation for square root of complex number

How do I represent $\sqrt{1 + ja}$? I'm trying to show that it's approximately equal to $\pm(1 + \frac{ja}{2})$ when $a \leq 1$.
1
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2answers
23 views

Proving the well ordering principle

THe well ordering principle has that every subset of $\mathbb{Z}^+_0$ has a least element. or if $S$ is a non-empty subset of $\mathbb{Z}^+_0$ and $S = \{a_1, a_2, a_3 ... a_n\}$, then there is a ...
0
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0answers
29 views

Show that $\lnot\exists x\in A(P(x))$ is equivalent to $\forall x\in A(\lnot P(x))$

This exercise is from "How to Prove it" by Daniel J Velleman. Chapter 2.2 #5 Show that $\lnot\exists x\in A(P(x))$ is equivalent to $\forall x\in A(\lnot P(x))$
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1answer
41 views

Contradiction proofs

I'm a first year Physics student and I have some trouble approaching Proofs by Contradiction in some of my Math classes. Once I get the first 2 or 3 statements I can finish the proof but a lot of the ...
2
votes
1answer
35 views

Proof Norm is Continuous

Someone just asked me why the norm of a normed space is continuous, and the answer I gave them satisfied them, but I'm not sure if it should. Something seems amiss. Let $\rho: X \to \mathbb{R}^+_0$ ...
1
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1answer
57 views

Finding the characteristic polynomial of $A^2$ given the characteristic polynomial of $A$

To find the characteristic polynomial of the matrix $A^2$, would I just compute $$(\lambda^2+4\lambda-5)^2 ?$$
2
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1answer
26 views

Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
0
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1answer
16 views

For a matrix $O$ containing columns which are an orthonormal basis for a column space, why does $O^{\prime}O = I$?

Theorem: let $\{o_i\}_{i \in \{1, 2, \dots, r\}}$ be an orthonormal basis for the column space of a matrix $X$ and let $O = \begin{bmatrix}o_1 & o_2 & \cdots & o_r\end{bmatrix}$. Then ...
-2
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4answers
32 views

Proof $log_{r} a = log_r s \cdot log_s a $ [on hold]

Do you know any proof of this logarithms property: $log_{r} a = log_r s \cdot log_s a $
0
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1answer
52 views

Mathematical Induction. Horses made me question my understanding

I recently read about the false inductive proof that all horses are the same colour. There are some mathSE threads about this already (MathSE_thread_1, MathSE_thread_2). After reading this, I now ...
2
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0answers
26 views

Does there exists a positive $t$ that satisfy this given condition?

I am curious about the validity of my claim concerning the equations: $(2k-1)t+1$ (1) $(2k^2-2k)t+(2k-1)$ (2) where $k=2,3,4,...$ My claim is for almost all $k$ or for infinitely many $k$, there ...
0
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1answer
24 views

$A$ is positive definite if and only if $Q$ is invertible for every choice of $Q$

Note that if $A \in M_{n \times n}$, $A^{\prime}$ denotes the transpose of $A$. I proved the following theorem already: $A$ is nonnegative definite if and only if there exists a square matrix ...
4
votes
4answers
61 views

Proving that $\varphi(n)=n\prod (1-1/p)$ without using multiplicativity

$$\varphi(n)=n\prod_{p \ \text{prime}} (1-1/p)$$ Can this useful formula be derived without using the fact that Euler's totient function is multiplicative?
2
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5answers
46 views

$A\backslash (B\cap C) = (A\backslash B)\cup (A\backslash C)$; only one inclusion seems to work

I encountered the following problem: $$A\backslash (B \cap C) = (A\backslash B)\cup(A\backslash C).$$ So I need to prove two things: $A\backslash (B\cap C) \subseteq (A\backslash B) \cup ...
0
votes
1answer
23 views

Map between two metric spaces and their limit

I have to proof the following: Let $f: V \to W$ be a map between two metric spaces. Proof that $f(a)$ with $a \in V$ is continuous if and only if, for every converging sequence $x_n$ in V with limit ...
0
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1answer
15 views

Finding the upper boundary of a sequence to find it's limit

I have to determine if the following sequences converge and if they converge I have to determine to what they converge and proof this. $$ a_n = 2^{-n} \\ b_n = \frac{n^2}{n^3 -10} \\ c_n = 1 ...
0
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1answer
26 views

Could the Hamel basis of $\mathbb{R^Z}$ be the set $\mathbb{R^Z}-{\mathbf{\{0\}}}$?

This is the follow up question to this question (*) According to page 2 of this link 1 and this link 2, $\mathbb{R^Z}$ (which is referred as $\mathbb{R^\infty}$ in link 1) has elements of the ...
2
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1answer
32 views

Why this set is the pole set of $z$?

Suppose $V\subset \mathbb P$ a projective variety, $z$ a rational function on $V$ and let's define $J_z=\{F\in k[X_1,\ldots, K_{n+1}\mid \overline Fz\in \Gamma_h(V)\}$ and $S_z$ the polo set of $z$, ...
0
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2answers
15 views

Let $C(X)$ be the column space of $X \in M_{n \times p}$. Starting off the proof of $C(X) \cup C(X)^{\perp} = \mathbb{R}^n$

Let $C(X)$ be the column space of $X \in M_{n \times p}$. Prove or disprove the following statement: Every vector in $\mathbb{R}^n$ is in either $C(X)$ or $C(X)^{\perp}$ or both. I interpret ...
-3
votes
2answers
40 views

How should I go about this proof? [on hold]

Let $a,b > 0$ be real numbers. Prove that $2ab \leq (a+b)\sqrt{ab}$. I'm new to proofs and would like some help understanding how to approach this proof. Thank You.
4
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0answers
18 views

Why is this a corollary of this theorem?

Lang - Algebra p.251 Proposition 6.11 Let $E/F$ be a normal field extension. Let $E^G$ be the fixed field of $Aut(E/F)$. Then, $E^G$ is purely inseparable over $F$ and $E$ is separable over ...
2
votes
4answers
67 views

Is there a book on proofs with solutions?

I am a biochemistry graduate student who works on cancer. I am interested in learning proofs as a personal interest. I use math as a tool, but would like to start building a deeper understanding on my ...
2
votes
2answers
54 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
13
votes
7answers
157 views

What are statements about the natural numbers where induction is impossible or unnecessary to prove?

I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, ...
2
votes
1answer
44 views

I need help in this proof of this exercise from Fulton's book

I'm reading Fulton's algebraic curves book. I'm trying to understand this solution which I found online of the question 4.17 on page 97. What I didn't understand is why $V(J_z)$ are exactly those ...
0
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1answer
27 views

The Change-making problem algorithm proof (at the dynamic programming method)

I saw here the algorithm for the "Change-making problem" (at the dynamic programming method). I saw it here: http://www.columbia.edu/~cs2035/courses/csor4231.F07/dynamic.pdf I'm trying to find a ...
0
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0answers
24 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
2
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1answer
36 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
4
votes
1answer
42 views

Proof in Fulton's *Algebraic Curves*

I'm reading Fulton's algebraic curves book on page 106 and I didn't understand this proof: I didn't understand why can we assume $F_Y\neq 0$? (what $F$ irreducible has to do with this?). ...
1
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2answers
31 views

Leibniz Formula, proof of alternating property

$$F_{A} := \sum_{\sigma\in S_n}\operatorname{sign}(\sigma) \prod_{i=1}^n A_{i \sigma(i)}$$ I am trying the prove that $\det(A)=F(A)$. I know that to do this, I need to show that $F$ satisfies the ...
0
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1answer
29 views

Proving that $f(x)=0\ \forall x\in B(0,r)$

Let $y=f(x_1,…,x_n)$ be differentiable on $B(0,r)$. Assume that $\dfrac{\partial}{\partial x_i}f(x)=0\ \forall x\in B(0,r)$ and $i\in\{1,…,n\}$. How to prove that $f(x)=0\ \forall x\in B(0,r)$? Do ...
1
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1answer
23 views

Proving that the ball is converx

I need to prove that the ball $B(x,r)=\{y\in \mathbb{R^n}:||y-x||<r\}$ is convex. How to do this?
1
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1answer
36 views

Help to understand this proof in Fulton's book

I'm reading Fulton's algebraic curves book on page 105 and I didn't understand this proof: 1.Why if $R=k[X_1,\ldots,X_n]$, then $\Omega_k(R)$ is generated (as R-módule) by the differentials ...
0
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1answer
60 views

Proving set identities

I am attempting to work on some proofs for my math assignment, but I'll be honest in that I am really struggling to understand them. I read through the power point given by my teacher; however, even ...
1
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4answers
83 views

Given a specific trapezoid, prove it is a rectangle

In quadrilateral ABCD, AB is parallel to CD. AC and BD meet at E. Points M and N are the midpoints of AE and DE, respectively. BM and BE trisect angle ABC, and CE and CN trisect angle BCD. Prove ...
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2answers
39 views

proof of laplacian $1/\rho$ in cylindrical coordinates at $\rho = 0$

In spherical coordinates, it can be proved that the laplacian of $r = \sqrt{x^2+y^2+z^2}$ at the origin is \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi\delta(\vec{r}) \end{align} as demonstrated in ...
0
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1answer
14 views

Heptagon (septagon) diagonal intersection

Diagonals TV and UW of regular heptagon TUVWXYZ meet at A. Prove that TU+TA=TW (Source: AoPS ItG). My observations: TUVW is an isosceles trapezoid Triangle TUA is congruent to triangle WVA
1
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2answers
50 views

Find the lowest degree of the polynom $P$?

I have to determine the lowest degree of $P$ given by the following system : $\left\{ \begin{array}{l} P \equiv 2X \ \mod[X^2 -2X +1] \\ P \equiv 3X \ \mod[X^2 -4X+4] \end{array} \right.$ First, ...
1
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2answers
19 views

Will decreasing the variance of a subset, the global variance also decrease and vice versa? [closed]

While implementing one of our propose algorithm we are assuming that, by decreasing the variance of a subset, the global variance will also decrease and vice versa. Is it something we need to proof ...
2
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1answer
25 views

Partial mapping, how does the inverse look like?

Let $X_1,X_2,Y$ be topological spaces. Let $f:X_1\times X_2 \to Y$ be continous at $a=(a_1,a_2)$. Show that the partial mappings $f_1:X_1\to Y; x\mapsto f_1(x) = f(x,a_2)$ is continous at $a_1$ and ...
4
votes
2answers
48 views

Prove Lagrange's Identity without induction

Prove Lagrange's Identity without induction. $$ \sum_{1\leq j <k\leq n}(a_jb_k-a_kb_j)^2=\left( \sum_{k=1}^na_k^2 \right)\left( \sum_{k=1}^n b_k^2 \right)-\left( \sum_{k=1}^na_kb_k \right)^2 $$ I ...
0
votes
1answer
39 views

Help me with this solution of the exercise 4.17 from Fulton's Algebraic Curves

I'm studying Fulton's algebraic curves book and in order to prove the well-definiteness of the divisor $div(z)$ on page 97 I'm trying to understand this solution which I found online. I didn't ...
4
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1answer
29 views

Prove: If $(\exists V $ neighbourhood of $ x)(f_{\mid V}:V\to Y$ is continous) then $f$ is continous in $x$.

Let $f: (X,\tau_X) \to (Y,\tau_Y), x\in X$ and $v\subseteq X$; Prove: If $(\exists V $ neighbourhood of $ x)(f_{\mid V}:V\to Y$ is continous in $x$) then $f$ is continous in $x$. I get ...
0
votes
0answers
17 views

Approximation of large unitary matrices

I have a hunch, that for now is only motivated by a physical argument. Could anyone point me in the right direction for proving (or disproving) it? The hunch is as follows: a unitary matrix $U$ of ...
0
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2answers
40 views

Finite fields, characteristics and the Fundamental Homomorphism Theorem

I am trying to make sense of this proposition. I am fine with part (a), for part (b) however, can you explain what the computation proves? Can you not verify a homomorphism by checking the 3 ...
1
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1answer
17 views

Can I assume a condition in the consequent?

Im reading Axler's Linear Algebra Done Right. In an exercise, he ask to prove that $$a\in F,v\in V,av=0 \implies a=0 \lor v=0 $$ where $V$ is a vector space over the field $F$. I've proved it this ...