# All Questions

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### Orthonormal Basis and Matrix of a Linear Operator Proof.

Let B = $\{v_1, \dots, v_n\}$ be an orthonormal basis for Rn Let p = $[v_1, \dots, v_n$]. Prove that for any x, we have that the B-matrix of x is equal to the tranpose of P times x. I am unsure as to ...
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### Maximal value of $\vert r^2-n\vert$ with a special condition

let $M,n\in \mathbb{N}$ and $R=\lbrace r\in \mathbb{N} \mid \vert r- \sqrt{n}\vert <M<2\sqrt{n}\rbrace$. I have to show that the maximal value of $\vert r^2-n\vert$ for $r\in R$ is at most ...
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### Prove that this extension is Galois with galois group the quaternions

Let $M=\mathbb Q(\sqrt{2},\sqrt{3})$ and let $E=M(\sqrt{(\sqrt{2}+2)(\sqrt{3}+3)})$. Prove that: M is Galois over $\mathbb{Q}$ Show that $E$ is Galois over $\mathbb Q$ with Galois group the ...
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### Determinant Question

This is not a homework question. Let $A$ be an $n\times{n}$ matrix with entries in the field $F$ such that each entry is relatively prime to any other entry. What is the number of elementary matrices ...
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### Implicit Differentiation and Tangent Lines (sin and cos)

a. verify that the given point lies on the curve. b. determine an equation of the line tangent to the curve at the given point function : $\cos{(x-y)}+\sin y = \sqrt{2}$ a. I derived it using ...
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### Chapter II Example 3.3 -p.28 Silverman

I have a question about Chapter II Example 3.3 -p.28 in Silverman "Arithmetic of Elliptic Curves". I feel like I'm misreading it and would like clarification. Let $K$ be a field such that ...
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### Is the projection of two isomorphic direct sum of modules isomorphic?

If A and B are R-modules, and if A x A is isomorphic to B x B, does this imply that A is isomorphic to B?
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### question about vector spaces and linear algebra

LEt $V = \{ (x,y) : x,y \in \mathbb{C} \}$. The addition and scalar multiplication are the usual ones. We know $V$ is a vector space over $\mathbb{C}$ with dimension $2$. My question: Why does $V$ is ...
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### Area between $y = \frac{2}{1+x^2}$ and $y = |x|$

The exercise is from Anton 8th, page 443, question 15. The problem asks for the area between: $f(x) = \frac{2}{1+x^2}$ and $g(x) = |x|$ It is not said what interval the area should be calculated, so ...
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### Intersection of Level Curves and a Ellipse at a given angle

I am preparing for an exam and I'm going over previous administered tests. I have come across the following problem and have little idea how to tackle it. It goes as follows: Let ...
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### SELF COMPLEMENTARY GRPAHS [on hold]

A simple graph is called self-complementary if it is isomorphic to its own complement. Let G be a simple graph on n vertices. Prove that if G is self-complementary, then either n = 4t or n = 4t + 1 ...
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### Using recursion tree to solve recurence T(n) = 3(n/2)+n

I am trying to solve the recurrance of the function, T(n) = 3(n/2)+n where T(1) = 1 and show it's time complexity. n can be ...
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### So why isn't $\Bbb R^n = \oplus _{n = 1}^{m}\Bbb R^n$

I thought that direct sum means each component of $V = \oplus U_i$ can be decomposed into elements of $U_i$. But if $U_i$ is replaced by the whole space, doesn't it mean the everything else in the ...
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### Let G be a group and let x be a fixed element of G. Define Γ(x) = {g ∈ G : gx = xg} [on hold]

(a) Prove that Γ(x) is a subgroup of G. (b) Let G = A4, let x = (1 3)(2 4) and let y = (2 4 3). Find (i) Γ(x) and (ii) Γ(y).
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### A Contradiction of Riemann Zeta Residues

We can show (1+2+3...+n)^2 = 1^3 + 2^3 + ... +n^3, which holds for any finite n, shouldn't this imply Z(-1)^2 = Z(-3)? However, this does not hold if we look at the residues of the zeta function ...
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### Translate the following sentence into conjunctive normal form

"Anyone who has cats as pets will not have mice": $$\forall x[\exists zHave(x,cat(z))]\rightarrow \forall y[\neg Have(x,mouse(y))]$$ I need to translate this into conjunctive normal form. So the ...