# Questions tagged [solution-verification]

For posts looking for feedback or verification of a proposed solution.

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### Fermat' Last Theorem

Fermat‘s Last Theorem Fermat‘s last theorem (proofed by Andrew Wiles in 1994) a^(i) – b^(i) <> c^(i) a, b, c, i are elements of N with a > b, i >1, i<>2; a, b, c are coprime. I) We ...
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### Any matrix has a unique decomposition of a sum of a symmetric and anti symmetric matrix proof

I am given the following task: Any $n \times n$ matrix $\ A$ can always be written,$$A= S+ C$$ Where $S$ is symmetric and $C$ is antisymmetric. Prove that this decomposition is unique. My ...
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### Find $lim _{n\to \infty}\frac{n-1}{n-2}$

I'm taking a university real analysis course and I have been tasked with proving that the sequence $x_n = \frac{n-1}{n-2}$ converges using first principles. First fix $\epsilon >0$. Using the ...
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### If A is a square matrix, and A² = A, prove that the column space C(A) = { x ∊ Rⁿ | x = Ax }

Here is my attempt at the proof. See that A can be written out as ...
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### Prove that $|x|<|y| \iff x^2<y^2$.

My proof is below. I am not sure about the second part. Proof: First I prove $|x|<|y| \implies x^2<y^2$. Suppose $|x|<|y|$. Then, \begin{align*} |x||x| < |y||x| &\implies x^2<|y||x|\...
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### Convert $\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$ and $\ddot{y}(t) = \dot{y}(t) -y(t)^2$ into 1. Order IVP.

1) We're given the IVP 3. Order $$\dddot{y}(t) = (\dot{y}(t)- y(t))^2 + 3\sin{(t)}y(t)$$ with initial values $y(0)=a, \dot{y}(0)=b, \ddot{y}(0)=c$ and want to convert it into a 1. Order IVP. 2) We're ...
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### Find $c(x,y), d(x,y) \in F[x,y]$ such that $a(x,y)c(x,y) + b(x,y)d(x,y)$ is a non-zero polynomial in $F[X]$

Let $F$ be a field. Let $a(x,y), b(x,y)$ be co-prime in $F[x,y]$.Prove that there exist $c(x,y), d(x,y) \in F[x,y]$ such that $a(x,y)c(x,y) + b(x,y)d(x,y)$ is a non-zero polynomial in $F[x]$. Here’s ...
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### How to prove that for any set E, the Lebesgue outer measure of E is bounded by Jordan inner and outer content?

Prove that for any $E \subset \mathbb{R}^2$ $c_i (E) \leq m∗ (E) \leq c_e (E)$. Thus, when E is rectifiable we get $m∗(E)=c(E)$. where $c_i$ is Jordan's interior content and $c_e$ is Jordan's exterior ...
### Prove that any two simple extensions of $\Bbb{R}$ are isomorphic (hence isomorphic to $\Bbb{C}$)
This question originates from Pinter's Abstract Algebra, Chapter 27, Exercise F5. If $a$ and $b$ are nonsquares in $\Bbb{R}, a/b$ is a square (why?). Use the same argument as in Exercise F4 to ...
There are two pendulums hanging from the same point, having length $l$. The pendulum bob has a diameter $d$. The planes of oscillations of the two pendulums are perpendicular to each other. One of ...