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awarded  Popular Question
Apr
21
comment For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?
Thank you. I get it.
Apr
20
comment For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?
@EwanDelanoy. I have two more quick questions. First, what did you mean about the notation $f$ in the last paragraph? Did it mean the transformation $T$? Second, why $\mathbb R^n=\text{Ker}((f−\lambda id)^{s_1})$? Please confirm.
Apr
18
comment Determinant of a matrix with specific main diagonal
Thanks Chappers for your detailed answer. I actually found the same to your last expression. The result I used at the beginning was a modification for that. It's good to hear I ended up with the correct answer.
Apr
18
accepted Determinant of a matrix with specific main diagonal
Apr
18
asked Determinant of a matrix with specific main diagonal
Apr
17
comment A question about unitary block matrix
Thank you Davide Giraudo.
Apr
17
accepted A question about unitary block matrix
Apr
16
asked A question about unitary block matrix
Apr
6
comment For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?
@EwanDelanoy Thank you. I had a bit different exaplanation for that. Set $T=f-\lambda{\textsf{id}}$ and we wanna show that $W_{p+1} \subset W_1$ with the hypothesis induction that $W_p=W_1, \forall p \ge 1.$ Let $v \in W_{p+1} \implies T^{p+1}(v)=0\implies T(v) \in W_p=W_1$. Therefore, $T^2(v)=0$ hence $v \in W_2=W_1$. In short, $v \in W_1$. This proves the claim.
Apr
5
comment For $n \ge 2$ , does every linear operator on $\mathbb R^n$ has an invariant subspace of dimension $2$ ?
@EwanDelanoy Can you explain why ${\sf Ker}((f-\lambda{\textsf{id}})^p)=W_1$ for every $p$? I understood that by induction we need to prove that $W_p=W_1$. The direction $W_1 \subset W_p$ is easy to prove. How about $W_p \subset W_1$?
Jan
27
accepted Example of continuous function over $\mathbb R^n$
Jan
27
revised Example of continuous function over $\mathbb R^n$
added 1 character in body
Jan
26
comment Example of continuous function over $\mathbb R^n$
$\epsilon-\delta-$ proof is used for real-valued functions so I think it couldn't applied for my case.
Jan
26
asked Example of continuous function over $\mathbb R^n$
Jan
26
comment Intersection of “positive” open half-spaces
Thank you @copper.hat. I got it.
Jan
26
accepted Intersection of “positive” open half-spaces
Jan
26
comment Intersection of “positive” open half-spaces
Dear @copper.hat. Can you give me the explanation why the equation $[x_1^T \ldots x_n^T]^T a = [1 \ldots 1]^T$ always has solution?. This would be a very nice solution for me.
Jan
26
comment Intersection of “positive” open half-spaces
Thank you @copper.hat. If we assume that $V=\mathbb R^n$ then your terminologies "$(\cdot, \cdot)$ is the inner product on $V$" and "the latter inner product is the standard inner product on $\mathbb{R}^n$" are identical isn't it?
Jan
26
comment Intersection of “positive” open half-spaces
Thank you @copper.hat. Since your notation had a bit different to mine so I would like to address some questions. I suppose so $V=\mathbb R^n.$ What did your notations $y_k,x_k,e_k$ mean? And I don't really get the point of "Define an inner product on $\mathbb R^2$ by $\langle a,b \rangle_V = (\phi(a),\phi(b) )$". Why $\mathbb R^2$ here? Thank you.