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Tutor profile: Andres Q.

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Andres Q.
College Math Teacher for 2 years
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Questions

Subject: Linear Algebra

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Question:

Let $$T(x,y,z) = (x+y, y+z,z)$$ ( a linear transformation) and $$W= T-I$$. Which is the dimension of the kernel space of $$W^3.$$

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Andres Q.
Answer:

$$dim(Nu(W^3))=3$$ because $$W(x,y,z) = (x+y,y+z,z)-(x,y,z) =(y,z,0)$$ then: $$W^3(x,y,z)= W^2(W(x,y,z))=W^2(y,z,0)= W(W(y,z,0))= W(z,0,0)= (0,0,0)$$ and that is the null function on $$R^3$$.

Subject: Trigonometry

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Question:

Is a rotation of a circle $$S$$ of radius $$r$$ an isometry?

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Andres Q.
Answer:

Yes, as $$S$$ is homeomorphic to $$S_1$$ we can try out the problem in here. If $$d$$ is the metric of the euclidian space in which the circle $$S_1$$ is in, and $$\theta$$ is the angle of rotation, then to any pair of points $$(cos\alpha, sin\alpha) , (cos\beta, sin\beta)$$on the circle we have $$d(f((cos\alpha, sin\alpha)), f((cos\beta, sin\beta))) = d((cos(\alpha+\theta), sin(\alpha+ \theta)), (cos(\beta+\theta), sin(\beta+\theta))= d((cos\alpha, sin\alpha), (cos\beta, sin\beta)).$$

Subject: Set Theory

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Question:

Let $$\beta\mathbb{N}$$ the set of ultrafilter on the natural numbers. Prove that there exists an ultrafilter $$p \in \bigcap_{n \in \mathbb{N}}^{}{(4n\mathbb{N}+2n)^* }$$ .

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Andres Q.
Answer:

Note that is equivalent to prove than the set $$\bigcap_{i=1}^{k}{(4n_i\mathbb{N}+2n_i) }\neq \emptyset$$ for a finite amount of naturals $$\left\{{n_i}\right\}_{I=1}^k$$, but this is true because for any natural $$j$$ we have that $$x=4j\prod_{i=1}^{k}{n_i} +2\prod_{i=1}^{k}{n_i} \in \bigcap_{i=1}^{k}{(4n_i\mathbb{N}+2n_i) }$$. QED

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