Difference between revisions of "Talk:Aufgaben:Problem 13"
(alternative proof of (a)) |
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It seems to be shorter, and thus is probably better for the exam... [[User:Carl|Carl]] ([[User talk:Carl|talk]]) 23:16, 13 June 2015 (CEST) | It seems to be shorter, and thus is probably better for the exam... [[User:Carl|Carl]] ([[User talk:Carl|talk]]) 23:16, 13 June 2015 (CEST) | ||
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| + | There is still one problem with the proof of part (b). We are supposed to show that for every \( s \in {1,...,k}\) there is a nontrivial invariant subspace \( W \subset V_s\) if you look closely I don't actually prove this. I included the case where \(dim V_s = 1 \Rightarrow V_s = W\). But this case seem to be excluded by the question where it says that W has to be non-trivial, if non-trivial means, what i think it means: \(W \neq \{0\}, V_s\). So there should be a reason why \(V_s\) is always reducible. Anyone have an idea for this? Best, | ||
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| + | [[User:Carl|Carl]] ([[User talk:Carl|talk]]) 10:05, 14 June 2015 (CEST) | ||
Revision as of 08:05, 14 June 2015
There is an alternative way of proving (a) in the Felder Script:
https://people.math.ethz.ch/~felder/mmp/mmp2/
see chapter 3 -> Satz 3.1. and chapter 2 -> Satz 2.6 for a prove of Schur's lemma.
It seems to be shorter, and thus is probably better for the exam... Carl (talk) 23:16, 13 June 2015 (CEST)
There is still one problem with the proof of part (b). We are supposed to show that for every \( s \in {1,...,k}\) there is a nontrivial invariant subspace \( W \subset V_s\) if you look closely I don't actually prove this. I included the case where \(dim V_s = 1 \Rightarrow V_s = W\). But this case seem to be excluded by the question where it says that W has to be non-trivial, if non-trivial means, what i think it means: \(W \neq \{0\}, V_s\). So there should be a reason why \(V_s\) is always reducible. Anyone have an idea for this? Best,
