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### Exercise 10, Section 3.4 of Hoffman’s Linear Algebra

If you're familiar with the idea of diagonalization, then since $S$ satisfies $S^2=S$ and is an operator on $\mathbb{R}^2$, the characteristic polynomial of $S$ is then $p(z)=z^2-z$. Since $S\neq I$ ...
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### Solution Verification: An equivalence relation on finite subsets of $\mathbb{N}$, defined by having equal sums of the elements in the sets

Your solution to $(1)$ neglects something: that $B$ is just the set $\{2,2,2\}=\{2\}$; this is why you should be more careful and write things explicitly as sets. (We delete duplicate entries in sets.)...
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### Convex $n$-gons that can be decomposed into $2n$ right triangles

I assume by "rectangle triangle" you mean "right triangle". Take a convex $n$-gon $A$ with consecutive vertices $V_i$ for $1\leq i\leq n$. Fix $P=V_n$. Form $n-2$ (in general non-...
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### Correctness of the proof that $\lim_{n \to \infty}(x_{n}-x_{n-2})=0$ implies $\lim_{n \to \infty}\frac{x_{n}}{n}=0$.

If $n$ is odd numbers,we have \begin{align} \lim_{k \to \infty} \frac{x_{2k+1}-x_{2k-1}}{2k+1-(2k-1)}=\lim_{k \to \infty} \frac{x_{2k+1}-x_{2k-1}}2&=0\\ \\ \Rightarrow \lim_{k \to \infty} \...
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### Applying a given formula to induction step.

I assume that you meant to write: $$x_{n+1} = \sqrt{2x_n-1}$$ The square root looks a bit weird in your post but I'm guessing that this is what you wanted. So, we have that: x_{n+1}-x_n = \sqrt{2x_n-...
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### Is this proof that one can pull an existential quantification out of a universal quantification using a cartesian product of the power set correct?

The proof does look correct. And it doesn't require the Axiom of Choice. In fact, it constructs exactly the sort of set one might then apply the Axiom of Choice on as a next step.
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Expanding on a way from my hint: You noted if $S \neq 0$ and $S \neq I$, we must have vectors $\alpha_1, \alpha_2 \in \mathbb{R}^2$ with $S \alpha_1 \neq 0$ and $S \alpha_2 \neq \alpha_2$. To satisfy $... • 5,444 1 vote Accepted ### Matchstick dice game winning strategy We can describe the current game state as$(n,d)$where$n \in \mathbb{N}$(including zero) is the number of nuts remaining and$d \in \{1,2,3,4,5,6\}$is the number on the top face of the die, ... • 5,444 1 vote Accepted ### Exercise 11, Section 3.5 of Hoffman’s Linear Algebra Since$W_1\cap W_2$is a subspace of$W_2$, there is a complementary subspace$U_2$such that$W_2=(W_1\cap W_2)\oplus U_2$. Let$U$be a complement of$W_1+W_2$in$V$, So$U\oplus U$is a complement ... • 306 1 vote ### Prove that two sets A and B with$A \cap B=\emptyset$,$\sup A = \sup B$,$\sup A \notin A$and$\sup B \notin B$cannot exist. Let$A:=\left\{1-\frac{1}{2n-1}\mid n\in\left\{1,2,\dots\right\}\right\}.$Let$B:=\left\{1-\frac{1}{2n}\mid n\in\left\{1,2,\dots\right\}\right\}.\$
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