grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2010 centrale-maths2__mp

32 maths questions

QI.A.1 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
For every element $x$ of $E$, we denote by $h(x)$ the application from $E$ to $E$ such that $\forall y \in E, h(x)(y) = \varphi(x,y)$.
a) Show that, for all $x$ in $E$, $h(x)$ is an element of the dual of $E$, denoted $E^{*}$.
b) Show that $h$ is a linear application from $E$ to $E^{*}$.
QI.A.2 Proof Proof of Set Membership, Containment, or Structural Property View
If $A$ is a subset of $E$, we denote $A^{\perp\varphi} = \{ x \in E \mid \forall a \in A,\ \varphi(x,a) = 0 \}$. Show that $A^{\perp\varphi}$ is a vector subspace of $E$.
QI.A.3 Proof Proof of Equivalence or Logical Relationship Between Conditions View
We say that $\varphi$ is non-degenerate if and only if $E^{\perp\varphi} = \{0\}$.
Show that $\varphi$ is non-degenerate if and only if $h$ is an isomorphism.
QI.A.4 Matrices Bilinear and Symplectic Form Properties View
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$.
a) Show that the matrix of $h$ in the bases $e$ and $e^*$ is: $$\operatorname{mat}(h, e, e^*) = \left(\varphi(e_i, e_j)\right)_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}}$$ This latter matrix will also be called the matrix of $\varphi$ in the basis $e$ and denoted $\operatorname{mat}(\varphi, e)$.
b) Let $(x,y) \in E^2$. We denote by $X$ and $Y$ the column matrices whose coefficients are the components of $x$ and $y$ in the basis $e$. Show that $\varphi(x,y) = {}^t X \Omega Y$ where $\Omega = \operatorname{mat}(\varphi, e)$ and where ${}^t X$ denotes the row matrix obtained by transposing $X$.
QI.B.1 Proof Direct Proof of a Stated Identity or Equality View
Let $q \in Q(E)$.
Show that there exists a unique symmetric bilinear form on $E$, denoted $\varphi$, such that $q = q_\varphi$.
QI.B.2 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $q$ be a quadratic form on $E$. Let $E'$ be a second $\mathbb{K}$-vector space of dimension $n$, and let $q'$ be a quadratic form on $E'$.
We call an isometry from $(E,q)$ to $(E',q')$ any isomorphism $f$ from $E$ to $E'$ satisfying: for all $x \in E$, $q'(f(x)) = q(x)$. We will say that $(E,q)$ and $(E',q')$ are isometric if and only if there exists an isometry from $(E,q)$ to $(E',q')$.
Show that $(E,q)$ and $(E',q')$ are isometric if and only if there exists a basis $e$ of $E$ and a basis $e'$ of $E'$ such that $\operatorname{mat}(q,e) = \operatorname{mat}(q',e')$.
QI.B.3 Proof Proof That a Map Has a Specific Property View
Let $p \in \mathbb{N}^*$. We denote by $c = (c_1, \ldots, c_{2p})$ the canonical basis of $\mathbb{K}^{2p}$. $$\text{For all } x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{K}^{2p}, \text{ we set } q_p(x) = 2\sum_{i=1}^{p} x_i x_{i+p}.$$
a) Show that $q_p$ is a quadratic form on $\mathbb{K}^{2p}$ and compute $\operatorname{mat}(q_p, c)$.
b) We call an Artin space (or artinian space) of dimension $2p$ any pair $(F,q)$, where $F$ is a $\mathbb{K}$-vector space of dimension $2p$, and where $q$ is a quadratic form on $F$ such that $(F,q)$ and $(\mathbb{K}^{2p}, q_p)$ are isometric. Show that in this case, $q$ is non-degenerate. When $p=1$, we say that $(F,q)$ is an artinian plane.
c) We assume that $\mathbb{K} = \mathbb{C}$ and for all $$x = \sum_{k=1}^{2p} x_k c_k \in \mathbb{C}^{2p}, \text{ we set } q(x) = \sum_{k=1}^{2p} x_k^2.$$ Show that $(\mathbb{C}^{2p}, q)$ is an Artin space.
d) We assume that $\mathbb{K} = \mathbb{R}$ and for all $$x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{R}^{2p}, \text{ we set } q'(x) = \sum_{i=1}^{p} x_i^2 - \sum_{i=p+1}^{2p} x_i^2.$$ Show that $(\mathbb{R}^{2p}, q')$ is an Artin space.
e) If $(F,q)$ is an Artin space of dimension $2p$, show that there exists a vector subspace $G$ of $F$ of dimension $p$ such that the restriction of $q$ to $G$ is identically zero.
QII.A.1 Proof Direct Proof of a Stated Identity or Equality View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We still denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$. Let $p \in \{1, \ldots, n\}$. We denote by $F$ the space spanned by $e_1, \ldots, e_p$.
a) Show that $F^\perp$ is the preimage under $h$ of $\operatorname{Vect}(e_{p+1}^*, \ldots, e_n^*)$, where $h$ is defined in I.A.1.
b) Show that $\operatorname{dim}(F) + \operatorname{dim}(F^\perp) = n$.
c) Show that $(F^\perp)^\perp = F$.
QII.A.2 Proof Direct Proof of a Stated Identity or Equality View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ and $G$ be two vector subspaces of $E$.
a) Show that $(F+G)^\perp = F^\perp \cap G^\perp$.
b) Show that $(F \cap G)^\perp = F^\perp + G^\perp$.
QII.A.3 Proof Proof of Equivalence or Logical Relationship Between Conditions View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a vector subspace of $E$. We denote by $\varphi_F$ the restriction of $\varphi$ to $F^2$. We will say that $F$ is singular if and only if $\varphi_F$ is degenerate.
Show that $F$ is non-singular if and only if one of the following properties is verified:
  • $F \cap F^\perp = \{0\}$;
  • $E = F \oplus F^\perp$;
  • $F^\perp$ is non-singular.
QII.A.4 Proof Proof of Set Membership, Containment, or Structural Property View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We say that two vector subspaces $F$ and $G$ of $E$ are orthogonal if and only if for all $(x,y) \in F \times G$, $\varphi(x,y) = 0$.
If $F$ and $G$ are two vector subspaces of $E$ that are orthogonal and non-singular, show that $F \oplus G$ is non-singular.
QII.B.1 Proof Computation of a Limit, Value, or Explicit Formula View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $E = \mathbb{R}^2$ and for all $(x,y) \in \mathbb{R}^2$, $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$.
Determine a $q$-orthogonal basis and a $q'$-orthogonal basis.
QII.B.2 Proof True/False Justification View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
With $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$ on $\mathbb{R}^2$ as defined in question II.B.1, does there exist a basis of $\mathbb{R}^2$ orthogonal for both $q$ and $q'$?
QII.B.3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We say that $e$ is $q$-orthogonal if and only if, for all $(i,j) \in \{1,\ldots,n\}^2$ with $i \neq j$, $\varphi(e_i, e_j) = 0$.
Suppose that $e$ is simultaneously $q$-orthogonal and $q'$-orthogonal. Show that, for all $i \in \{1,\ldots,n\}$, $e_i$ is an eigenvector of $h^{-1} \circ h'$.
QII.B.4 Invariant lines and eigenvalues and vectors Simultaneous diagonalization or commutant structure View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $h^{-1} \circ h'$ has $n$ distinct eigenvalues. Show that there exists a basis of $E$ orthogonal for both $q$ and $q'$.
QII.C.1 Proof Existence Proof View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $x \in E$ such that $q(x) = 0$ and such that $x \neq 0$.
We propose to demonstrate that there exists a plane $\Pi \subset E$ containing $x$ such that $(\Pi, q_{/\Pi})$ is an artinian plane (where $q_{/\Pi}$ denotes the restriction of the application $q$ to the plane $\Pi$).
a) Demonstrate that there exists $z \in E$ such that $\varphi(x,z) = 1$.
b) We set $y = z - \frac{q(z)}{2}x$. Compute $q(y)$.
c) Conclude.
QII.C.2 Proof Proof by Induction or Recursive Construction View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a singular vector subspace of $E$. We assume that $(e_1, \ldots, e_s)$ is a basis of $F \cap F^\perp$. We denote by $G$ a supplementary subspace of $F \cap F^\perp$ in $F$.
a) Show that $G$ is non-singular.
b) Demonstrate by induction on the dimension of $F \cap F^\perp$ (starting with $\operatorname{dim}(F \cap F^\perp) = 1$, then $\operatorname{dim}(F \cap F^\perp) > 1$) that there exist $s$ planes $P_1, \ldots, P_s$ of $E$ such that the following three properties are verified:
  1. For all $i \in \{1,\ldots,s\}$, $(P_i, q_{/P_i})$ is an artinian plane containing $e_i$.
  2. For all $(i,j) \in \{1,\ldots,s\}^2$ with $i \neq j$, $P_i$ is orthogonal to $P_j$.
  3. For all $i \in \{1,\ldots,s\}$, $P_i$ is orthogonal to $G$.
QII.C.3 Proof Proof of Set Membership, Containment, or Structural Property View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that $\bar{F} = G \oplus P_1 \oplus \ldots \oplus P_s$ is non-singular. We will say that $\bar{F}$ is a non-singular completion of $F$.
QII.C.4 Proof Direct Proof of an Inequality View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that if $q_{/F} = 0$, then $\operatorname{dim}(F) \leq \frac{n}{2}$.
QII.C.5 Proof Proof of Equivalence or Logical Relationship Between Conditions View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $n = 2p$. Show that $(E,q)$ is an Artin space if and only if there exists a vector subspace $F$ of $E$ of dimension $p$ such that $q_{/F} = 0$.
QIII.A.1 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself, that is, the set of automorphisms $f$ of $E$ satisfying: for all $x \in E$, $q(f(x)) = q(x)$.
Let $f$ be an endomorphism of $E$.
a) Show that $f \in O(E,q)$ if and only if, for all $(x,y) \in E^2$: $\varphi(f(x),f(y)) = \varphi(x,y)$. Show that if $F$ is a vector subspace of $E$ and if $f \in O(E,q)$, then $f(F^\perp) = (f(F))^\perp$.
b) Let $e$ be a basis of $E$. Compute the matrix of the bilinear form: $$(x,y) \mapsto \varphi(f(x),f(y)) \text{ in terms of } \operatorname{mat}(f,e) \text{ and } \operatorname{mat}(\varphi,e).$$
c) Let us set $M = \operatorname{mat}(f,e)$ and $\Omega = \operatorname{mat}(\varphi,e)$. Show that $f \in O(E,q)$ if and only if $\Omega = {}^t M \Omega M$.
d) Show that if $f \in O(E,q)$, then $\operatorname{det}(f) \in \{1,-1\}$. We will denote: $$O^+(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = 1\} \text{ and } O^-(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = -1\}.$$
QIII.A.2 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ and $G$ be two vector subspaces of $E$ such that $E = F \oplus G$. We denote by $s$ the symmetry with respect to $F$ parallel to $G$.
a) Show that $s \in O(E,q)$ if and only if $F$ and $G$ are orthogonal (for $\varphi$).
b) Deduce that the symmetries in $O(E,q)$ are the symmetries with respect to $F$ parallel to $F^\perp$, where $F$ is a non-singular subspace of $E$.
c) When $H$ is a non-singular hyperplane, we will call reflection along $H$ the symmetry with respect to $H$ parallel to $H^\perp$. Show that every reflection of $E$ is an element of $O^-(E,q)$.
d) Let $(x,y) \in E^2$ such that $q(x) = q(y)$ and $q(x-y) \neq 0$. We denote by $s$ the reflection along $H = \{x-y\}^\perp$. Show that $s(x) = y$.
QIII.B.1 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Suppose that $E$ is an artinian space of dimension $2p$ and that $F$ is a subspace of $E$ of dimension $p$ such that $q_{/F} = 0$.
If $f \in O(E,q)$ with $f(F) = F$, show that $f \in O^+(E,q)$.
QIII.B.2 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ be a subspace of $E$ such that $\bar{F} = E$ (where $\bar{F}$ is a non-singular completion of $F$). Show that if $f \in O(E,q)$ with $f_{/F} = \operatorname{Id}_F$ (where $\operatorname{Id}_F$ is the identity application from $F$ to $F$), then $f \in O^+(E,q)$.
QIII.B.3 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $f \in O(E,q)$. We assume that for all $x \in E$ such that $q(x) \neq 0$, we have $f(x) - x \neq 0$ and $q(f(x)-x) = 0$.
We propose to demonstrate that $f \in O^+(E,q)$ and that $E$ is an Artin space.
a) Show that $\operatorname{dim}(E) \geq 3$.
b) We denote by $V = \operatorname{Ker}(f - \operatorname{Id}_E)$. Show that $q_{/V} = 0$.
c) Let $x \in E$ such that $q(x) = 0$. We denote $H = \{x\}^\perp$. Show that $q_{/H}$ is not identically zero. Deduce that there exists $y \in E$ such that $q(x+y) = q(x-y) = q(y) \neq 0$.
d) We denote by $U = \operatorname{Im}(f - \operatorname{Id}_E)$. Show that $q_{/U} = 0$.
e) Show that $U^\perp = V = U$.
f) Deduce that $E$ is an Artin space and that $f \in O^+(E,q)$.
QIV.A.1 Proof Direct Proof of a Stated Identity or Equality View
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.''
Prove the Cartan-Dieudonné theorem when $n = 1$.
QIV.A.2 Proof Deduction or Consequence from Prior Results View
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 when there exists $x \in E$ such that $f(x) = x$ with $q(x) \neq 0$.
QIV.A.3 Proof Deduction or Consequence from Prior Results View
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 when there exists $x \in E$ such that $q(x) \neq 0$ and $q(f(x)-x) \neq 0$.
QIV.A.4 Proof Deduction or Consequence from Prior Results View
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).
QIV.B.2 Groups Symplectic and Orthogonal Group Properties View
We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') = 1$. Let $x \in F$ with $x \neq 0$. Set $y = f(x)$.
a) Show that $q(x+y)$ or $q(x-y)$ is non-zero.
b) Prove Witt's theorem in this case, using question III.A.2-d).
QIV.B.3 Groups Symplectic and Orthogonal Group Properties View
We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We now assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') > 1$.
a) Show that there exist $F_1$ and $F_2$ non-singular, such that $F_1 \perp F_2$ and $F = F_1 \oplus F_2$, with $\operatorname{dim}(F_1) = \operatorname{dim}(F) - 1$.
b) Suppose that there exists $g \in O(E,q)$ such that $g_{/F_1} = f_{/F_1}$. Denote $F_1' = f(F_1)$. Show that $f(F_2) \subset F_1'^\perp$ and that $g(F_2) \subset F_1'^\perp$.
c) Show that there exists $$h \in O\left(F_1'^\perp, q_{/F_1'^\perp}\right) \text{ such that } h_{/g(F_2)} = (f \circ g^{-1})_{/g(F_2)}.$$
d) Show that there exists $k \in O(E,q)$ such that $k_{/F} = f$.
QIV.B.4 Groups Symplectic and Orthogonal Group Properties View
We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
Prove Witt's theorem.