grandes-ecoles 2010 QIV.A.4

grandes-ecoles · France · centrale-maths2__mp Proof Deduction or Consequence from Prior Results

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude in the other cases (i.e., when neither of the conditions in IV.A.2 or IV.A.3 holds).

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''

We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.

Conclude in the other cases (i.e., when neither of the conditions in IV.A.2 or IV.A.3 holds).

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4