Hagen von Eitzen
Reputation
187,871
397/400 score
 12h comment How to prove that $\det A=|\lambda|^n$ You are right that $\dim V_0<\dim V$ implies that $A$ is singular. 12h comment Intersection of two circles giving reversed answer I don't see any coordinates reversed in the graphics on the referenced mathworld page 12h comment Why isn't the number of cosets equal to cardinalities of the groups? @mavavilj Yes. For example, if $x\in H$ (which happens for $|H|$ different $x$) we have $xH=H$. 21h comment Proving Injectivity $x + \sin(x)$ @ArturodonJuan That iniquality holds for all real $x\ne 0$. Even with the geometrical definition of sine, $\sin x$ is the straight line distance from the $x$-axis to a point on the unit circle, whereas $x$ is the not-so-straight length of the arc along the unit circle from the $x$-axis to that point 1d comment Let $\mathcal S$ be the collection of all straight lines in the plane $\mathbb R^2$. If $\mathcal S$ is a subbasis for a topology … @user290425 Of course, see also the meanwhile existing answer. 1d comment If $c$ is critical point and $x_{n}\to c$ then $f''(c)=0.$ Can we assume that $f$ is twice differentiable? Or only once and the existence of $f''(c)$ is to be shown? 1d comment Let $\mathcal S$ be the collection of all straight lines in the plane $\mathbb R^2$. If $\mathcal S$ is a subbasis for a topology … Actually, this new topology is also Hausdorff ... 1d comment In $\mathbb{Z}[\omega]$, if $a^3+b^3+c^3=0$ then $1-\omega$ divides at least one of $a,b,c$ @ThomasAndrews Your finding leads to $a\equiv b\equiv c\pmod{1-\omega}$. In case $\equiv 0$, we are done. In case $\equiv 2$, just mutiply everything with $2$ to arrive in the remaining case $a\equiv b\equiv c\equiv 1\pmod{1-\omega}$. Then $a^3+b^3+c^3\equiv 3\pmod 9$ by the useful result. 1d comment Under which additional hypothesis are open maps locally injective The constant map to a one-point space is open, but locally injective only at isolated points ... 1d comment Let $I=[0,1]$ and $f:[0,1]\to[0,1]$ be a continuous function. Then exist $x\in I$ such that $f(x)=x$. Consider $g(x)=f(x)-x$. Then $g(0)\ge 0\ge g(1)$ 2d comment Does there exist a function with following properties? I just see that after much reformulating my first attempt, this ends up as a filled-out version of @Siminore's answer - so maybe all credit should go there 2d comment Irreducibility of a Polynomial after a substitution +1 So here's how to show a polynomial is irreducible by successfully factoring it :) 2d comment Rational Question for $a + b$ and Irrationality of $a^2 + b^2$ ... where $\pi$ can be replaced with any irrational (except square roots of a rational) between $0$ and $\frac{10}3$ ... 2d comment Finding the Limit of a Sequence (No L'Hopital's Rule) +1 You even only need that $\lim_{h\to 0}(1+h)^{1/h}$ exists at all, not that it $e$. 2d comment Counting the number of “distinct” permutations of two sets? @Finalfire All possible $Y$ can be obtained as follows: Pick an arbitrary bijection $f\colon A\to B$. This induces a homomorphism $\phi\colon S_A\to S_B$, $\sigma\mapsto f\circ \sigma\circ f^{-1}$. Note that by this $(p,q)$ is equivalent to $(\sigma\circ p, \phi(\sigma)\circ q)$. Now pick an arbitrary map $g\colon S_A\to S_A$. Let $Y=Y_g=\{\,(g(p)\circ p,\phi(p))\mid p\in S_A\,\}$. - Hence there are $n!^{n!}$ possible $Y$ 2d comment Real numbers as element of a universe Why (in defining $\Bbb Z$ the usual way) do we need $-$ to write down $(a,b)\sim (c,d)\iff a+d=c+b$? This does use only $+$. - In contrast, idon't we need $-$ in order to define $+$ for $\{\,(a,b)\in\omega\times\omega\mid a=0\lor b=0\,\}$? 2d comment Prove: If $\ker(A)\cap \mbox{Im}(A)\not = \{0\}$ then $A$ is a singular matrix But already $\ker(A)\ne\{0\}$ implies that $A$ is singulkar ... 2d comment Prove $P(X=k \mid X+Y=n) = \frac1{n+1}$ Maybe as a guidance, the heuristic is: $X,Y$ can be viewed as "number of tails before first head" for a biased coin. We are looking for "number of tails before first head, given that there are $n$ tails before second head" or "given that exactly one out of $n+1$ flips is head, at which position does it occur?" - The latter variant is clearly symmetric in the $n+1$ positions, hence uniform. Apr 25 comment How to prove that Bezier(t) polynomial lies in convex hull of points (i/n,ai) for i from 1 to n "and i found that Bezier curve lies in convex hull of it's control points" - then what are the points $(i/n,a_i)$ if they are not the control points? Apr 25 comment Showing that the powers of a root of the p-th cyclotomic polynomial are distinct roots thereof. @Peter You do not use the (unstated by the OP) fact that $p$ is prime. For $p=4$, we might have $\gamma=-1$ and then fail because $\gamma^2$ is not a root of $x^3+x^2+x+1$ and because $\gamma^3$ and $\gamma^1$ are equal.