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Jan
25
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
(You might also find it helpful to note that the "natural" multiplication on $\mathbb{R}^2$, namely $(u,v)\cdot (x,y) = (ux, vy)$, does not agree with the natural multiplication on $\mathbb{C}$ for any choice of $g\in \mathrm{GL}_2(\mathbb{R})$.)
Jan
25
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
@laovultai: Let $f$ be the map $\mathbb{R}^2\to \mathbb{C}$, $(a,b) \mapsto a+ib$, and let $g$ be any element of $\mathrm{GL}_2(\mathbb{R})$. Then $f\circ g$ is an isomorphism $\mathbb{R}^2\to \mathbb{C}$ as $\mathbb{R}$-vector spaces. Now simply "pull back" the multiplication from $\mathbb{C}$ to $\mathbb{R}^2$ along the map $f\circ g$ (e.g. when $g$ is the identity map, the multiplication inherited is $(u,v)\cdot (x,y)=(ux−vy,uy+vx)$), and you get an isomorphism of rings (or, equivalently, $\mathbb{C}$-vector spaces). Does that answer your question?
Jan
17
comment How should I understand $R[x]/(f)$ for a ring $R$?
f should be irreducible, otherwise it is not true that R[alpha] = R[x]/f.
Jan
17
comment What's the difference between $\mathbb{R}^2$ and the complex plane?
@laovultai: Because they're not "equal". They're isomorphic, but in order to prove that, I have to choose an isomorphism. I chose the "obvious" one, $(a,b) \mapsto a+ib$. (There are lots more, e.g. $(a,b) \mapsto 2a + b - 5ia$, or $(a,b) \mapsto ia - b$.)
Aug
2
comment Surjection/Injection in Product of Linear Transformation
Also, why not just pick an explicit example? The first example I think of is $$S = \begin{pmatrix} 1&0&0\\0&1&0\\0&0&1\\0&0&0\end{pmatrix}, T = \begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}.$$ What properties does this have?
Aug
2
comment Surjection/Injection in Product of Linear Transformation
"Say, S(abc)=abcd, also S(abc)=pqrs." But this can never happen! Functions don't work like this. If you plug in an input, you get one output.
Jul
29
comment Should we simplify or not for function domain?
Do you mean "what is the domain of $f$"? Personally, if I was being very strict, I would say that the definition of $f$ was "take $x$, square it, and then divide the result by $x$", which always gives you $x$ back except when $x = 0$, when the calculation doesn't make sense. So the domain is the set of non-zero real numbers. (In practice, I am never this strict, except when I am teaching students how to be this strict.)
Jul
28
comment linear map vs operator: raised to power
Your example illustrates the point nicely, but of course, it's slightly worse than that: even if $V\cong W$, there's no obvious way of defining $T^2$. (Matrices are evil notation in that they hide the implicitly chosen bases. The OP might want to consider a map from $V$, the space of real polynomials of degree at most 4, and $W$, the space of real symmetric 3*3 matrices of trace 0. Both are $\cong \mathbb{R}^5$, and there are clearly lots of nice linear maps between them, but there's no obvious way of applying one twice.)
Jul
28
comment linear map vs operator: raised to power
What does $T^2$ mean, when $T: V\to W$? If I try to plug in a vector $v\in V$, I get a little confused: $T^2 v = T(T(v))$, but $T(v)\in W$, so I can't apply $T$ again.
Jul
28
comment what is the difference between sections and germs in a sheaf?
Sections are "functions on a large open set". Germs are "functions around a point". If this isn't obvious, why don't you tell us what your definitions are?
Jul
28
comment Fixed point of a continuous map
For the first problem, consider the map $g:x\mapsto f(x) - x$ on $[-1, 1]$. What is $g(0)$? What is $g(1)$? Where does $f$ have a fixed point?
Jul
28
comment Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$
@user1620696 No problem!
Jul
28
comment Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$
I've edited my post and added the surjectivity argument.
Jul
28
comment Find equation that represents $m$ as the subject
@Nick The answer is, essentially, that you need to plug numbers into formulas, or perhaps write a computer program to do so if you can't face doing it yourself. If you try to solve it algebraically, you'll end up with something disgusting like George's formulas above no matter how you spin it (Wikipedia's methods are all different ways of saying the same thing). I like George's recursive algorithm, but of course it (almost certainly) won't give you exact answers. Still, let me know if I can help with anything.
Jul
27
comment Find equation that represents $m$ as the subject
Perhaps the most appropriate tag is "algebra-precalculus". (Though I wouldn't expect anyone to learn how to solve a cubic equation until well after calculus, admittedly, but it has the same flavour as a quadratic equation, just much uglier...)
Jul
27
comment Why do I disagree with my calculator?
I agree with GEdgar, but if remembering that numbers have to be packaged with the sign to the left of them is hard, it's enough to remember that + and - are to be given the same priority, and so you should work from left to right, which always gives the right answer.
Jul
27
comment How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole
Ah, sorry - my three-dimensional mental picture confused me. Thanks for the clarification.
Jul
27
comment How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole
But all rotations that send P to S do arise in this way, right?
Jul
27
comment Euler's phi function $\phi(n)$ is even for all $n \geq 3$; when is it not divisible by $4$?
What do you know about $\phi$? Can you calculate, for example, $\phi(27)$?
Jul
27
comment Why do real powers need positive bases? (and to handle them)
You are absolutely right. It is not true that the power $a^b$ is only well-defined when $a\geq 0$. You can ascribe a well-defined value to $a^b$ for any $a$ when $b$ is a positive integer, for example; in fact, you can extend the function far more than that. But you lose a lot of its nice properties (e.g. $a^{b_1} a^{b_2} = a^{b_1+b_2}$) when you do so. It's fairly safe to say that the power $a^b$ is only defined *for all $b$* and *in a nice way* when $a\geq 0$.