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Aug
19
revised Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$
deleted amp;
Aug
19
comment Analytic functions on $\mathbb{H}$ such that $f(i)=3i$
Interesting question!
Aug
19
revised Analytic functions on $\mathbb{H}$ such that $f(i)=3i$
some formatting
Aug
19
comment If $f$ is one to one show that $f(a) \in \partial \Omega$
It looks ok to me. Where you write ...,so there exists a $\delta > 0$... etc. I would have written $F$ or $\widetilde{f}$ or something like this because it's an analytic extension of $f$ and not $f$ itself (as it is defined at $a$ and $f$ isn't).
Aug
19
revised If $f$ is one to one show that $f(a) \in \partial \Omega$
edited tags
Aug
19
comment Slick proof that if an open set contains $\mathbb Q$ it has all irrational numbers, except a countable amount.
@dREaM Oh I see, the "not countable" referred to the missing numbers.
Aug
19
comment Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
Nice answer. Where you write $n\alpha$ is in that interval didn't you mean $\operatorname{fractional part}(n\alpha)$ is in that interval?
Aug
19
answered $E$ is a certain subspace of $\mathbb{R}[x]$. Is the set $\{x − 2, (x − 2)^2, (x − 2)^3\}$ a basis of $E$?
Aug
19
comment Slick proof that if an open set contains $\mathbb Q$ it has all irrational numbers, except a countable amount.
@dREaM What is wrong with taking $\mathbb R$ minus $\{i_1, \dots, i_n\}$ where $i_k$ are irrationals?
Aug
18
answered Verify that Function is Homomorphism for $\mathbb{Z}_8 \rightarrow \mathbb{Z}_4$
Aug
18
revised A question about the proof of $(z_1z_2)^a=z_1^az_2^a$
edited title
Aug
18
comment Basis for a linear space
@DavidP Tru dat. I will not delete my comment though as I think it might be useful to others who read this thread.
Aug
18
comment Basis for a linear space
Showing that the set spans the space is not enough: one has to show that it's linearly independent, too.
Aug
14
awarded  Popular Question
Aug
10
awarded  Popular Question
Aug
9
comment Are all matrices linear operators?
But I would like to be convinced otherwise, of course.
Aug
9
comment Are all matrices linear operators?
In particular, if $A$ is such a matrix and $x$ some vector and we let $Ax$ be the usual multiplication then this seems to me is not a map $\{1,\dots,m\}\times\{1,\dots,n\} \to \mathbb{K}$ but a map $\mathbb R^m \to \mathbb R^n$.
Aug
9
comment Are all matrices linear operators?
I will think about this some time. For now I do understand what you meant in your previous comment but I still don't see why a matrix does not define a linear map.
Aug
9
comment Are all matrices linear operators?
Honestly, I really really don't understand how a matrix is a map $\{1,\dots,m\}\times\{1,\dots,n\} \to \mathbb{K}$. It seems to me that it takes vectors of arbitrary real numbers as argument not just integer pairs in these finite sets...?
Aug
9
comment Are all matrices linear operators?
An $n \times m$ matrix with real entries is a map $\mathbb R^m \to \mathbb R^n$. So please do continue.