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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
awarded  Custodian
Jan
17
reviewed Approve suggested edit on Optimization Word Problem
Jan
17
answered If $X$ and $Y$ are objects of $\mathrm{Set}$, is there any reason not to regard $\mathrm{Hom}(X,Y)$ as an object, too?
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 $(a,b) \mapsto ia - b$.)
Dec
22
awarded  Nice Answer
Sep
10
awarded  Yearling
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
29
awarded  Citizen Patrol
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
revised Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$
added 655 characters in body
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.