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1d
revised Show that $\phi(p^e)=p^e-p^{e-1}$
Fixed missing exponent
1d
comment Can an infinite sum of irrational numbers be rational?
So $S$ is rational, but different from $e^{\ln(x)} = x$, for any $x$ rational?
1d
comment Derived Algebra is nilpotent implies the lie algebra is solvable
$[L, L] = L^{(2)}$.
1d
answered Proving that $\binom{n}{k}\binom{\smash{k}}{m}\binom{m}{r} = \binom{n}{r}\binom{n-r}{n-m}\binom{n-m}{n-k}$
1d
comment subgroup having index $2$ of $R^*$
Of course ;-) $\ $
2d
comment subgroup having index $2$ of $R^*$
Not quite. $r^{2} H = H$ iff $r^{2} \in H$. So $H$ contains all squares. Now the set $S = (\mathbb{R}^{\star})^{2}$ of squares is a subgroup of index $2$, because $\mathbb{R}^{\star} = S \cup (-1)S$. So $H = S$.
2d
comment For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?
Because it sends each basis vector into a multiple of it.
2d
comment For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?
And then by the Lemma any matrix $A$ that commutes with $D$ has to send each of these eigenspaces to itself. Which means, $A$ is itself diagonal.
2d
comment For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?
Take $D$ to have distinct values on the main diagonal. What are the eigenvalues and the eigenspaces of $D$ then?
2d
revised linear algebra in infinite dimension
edited tags
2d
answered linear algebra in infinite dimension
2d
revised For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?
added 225 characters in body
2d
answered For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?
2d
revised Find $a,b,c \in \{1,2,..,9\}$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{10+a}{10+b}$
deleted 3 characters in body
2d
revised Find $a,b,c \in \{1,2,..,9\}$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{10+a}{10+b}$
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2d
answered subgroup having index $2$ of $R^*$
2d
comment Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$
@Jan, precisely!
2d
answered Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$
2d
revised Proper subgroup of $S_{15}$ that strictly contains $\sigma $
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2d
revised Proper subgroup of $S_{15}$ that strictly contains $\sigma $
added 117 characters in body