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Prove that for every real number $x$, if $x>2$ then there is a real number $y$ such that $y+1/y = x$.

I’m trying to learn about proofs and I’m stuck in Velleman’s book “How to Prove it”. This is the question (ex.6 p.118): Prove that for every real number $x$, if $x > 2$ then there is a real number $y$ such that $y+\frac{1}{y}= x$. I couldn’t do it so I went to see the answer, […]

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Prove set of matrices with eigenvalue $1$ is closed

The idea of this question is to find a continuous map onto a closed set, then take the preimage of that closed set to show that a particular set is closed. I’m stumped on this one question: Show the set of all $n \times n$ with all eigenvalues equal to $1$ is closed. What kinds […]

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Is it true that product of rank $n$ matrices is at most rank $n$?

The context is when doing $A^TA$ for a matrix $A \in M_{m,n}(\mathcal{R})$ say. Then $A^TA$ is invertible if column of $A$ are independent (I’m not sure about this fact either)?

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Symmetric difference – equality proofs.

I have a couple of statements to prove (self-learning, not homework). I’m not able to proceed with any of them (I’ve tried starting from RHS, LHS, etc.). I suppose I’m missing something. I would like to receive some tips or a solution for one of them and I will try to solve the rest. The […]

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Find the infinite product

Find the infinite product $$\left(\dfrac{2}{1}\cdot\dfrac{2}{3}\cdot\dfrac{4}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{5}\cdot\dfrac{6}{7}\cdot\dfrac{8}{7}\cdots\right)$$ I solved it, but my method is not nice one. I solved it using calculus (that’s why tagged). Please give a nice and sweet solution. Answer

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Transforming to polar coordinates

I want to transform from planar $x,y$ coordinates to $r,\theta$ coordinates, where $$r = \sqrt{x^2+y^2},\quad \theta = \arctan(y/x)$$ $$x = r\cos\theta, \quad y = r\sin\theta.$$ To do this I wish to compute the Jacobian matrix. One first step is computing $\partial r/\partial x$. I find $$\frac{\partial r}{\partial x} = (\frac{\partial x}{\partial r})^{-1} = (\frac{\partial{(r \cos\theta)}}{\partial […]

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Collinearity of two vectors.

Let $\vec{a}$ and $\vec{b}$ be two non-zero collinear vectors. $$\therefore x\vec{a}+y\vec{b}= \vec{0}$$ Now is it possible that either $x= 0$ or $y=0$?

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Equation of circumcircle of a triangle which has minimum area

From a point $ P(0,b) $ two tangents are drawn to the circle $ x^2+y^2=16 $ and these two tangents intersect x-axis at two points A and B .If the area of triangle PAB is minimum ,then prove that the equation of its circumcircle is $ x^2+y^2=32 $. The solution is given in my book […]

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A question about ideal

Suppose R is a ring. I is an ideal of R, A is a R-algebra, B is an ideal of A and $A/B\cong R$, can we prove $IB=IA\cap B$? I think it is true but can not express strictly.

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Generalizing the group law for elliptic curves

Let us consider the elliptic curve $C$: $$y² = x³ + ax² + b$$ where $a,b$ are integers. The formula for adding two points $(x₁, y₁)$ and $(x₂, y₂)$ in $C$ is expressed as a rational function of the coordinates $x₁, y₁,x₂, y₂$ (https://ocw.mit.edu/courses/mathematics/18-783-elliptic-curves-spring-2019/lecture-notes/MIT18_783S19_lec2.pdf), i.e., there exist four polynomials $w,s,r,d$ such that $$(x₁, y₁)+(x₂, y₂)=(w(x₁, […]