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.

Let $\alpha$ be a non-zero element of $E$. Show, for every vector $x$ of $E$, the identity: $$\tau _ { \alpha } ( x ) = x - 2 \frac { \langle \alpha , x \rangle } { \langle \alpha , \alpha \rangle } \alpha$$

In this question, the space $E$ has dimension $n \geq 2$. For every pair $(\alpha , \beta)$ of non-zero vectors of $E$, let $\theta _ { \alpha , \beta }$ be the geometric angle between $\alpha$ and $\beta$, that is, the unique element of $[ 0 , \pi ]$ given by: $\| \alpha \| . \| \beta \| \cos \theta _ { \alpha , \beta } = \langle \alpha , \beta \rangle$.
Let $\mathcal { R }$ be a root system of $E$ and let $\alpha , \beta$ be two non-collinear elements of $\mathcal { R }$.
a) Show, using property 4, that: $2 \frac { \| \alpha \| } { \| \beta \| } \left| \cos \theta _ { \alpha , \beta } \right| .2 \frac { \| \beta \| } { \| \alpha \| } \left| \cos \theta _ { \alpha , \beta } \right| \leq 3$.
b) Assume $\| \alpha \| \leq \| \beta \|$. Show that the pair $(\alpha , \beta)$ is found in one of the configurations listed in the table below (each row corresponding to a configuration):

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\| \beta \| / \| \alpha \|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

In this question, the space $E$ has dimension $n \geq 2$. Conversely, assume that a pair $(\alpha , \beta)$ of non-collinear vectors of $E$ is found in one of the configurations listed in the table below. Show that the real number $2 \frac { \langle \alpha , \beta \rangle } { \langle \alpha , \alpha \rangle }$ is an integer; specify its value.

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\| \beta \| / \| \alpha \|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

Show that $A \in \mathrm{SO}(2)$ if and only if there exists a real $t$ such that $A = R_t$ with $R_t = \left(\begin{array}{rr} \cos t & -\sin t \\ \sin t & \cos t \end{array}\right)$.

Write a procedure or function in Maple or Mathematica that takes as input a quadruple $(a,b,c,d)$ of reals and returns, when possible, a real $t$ such that $\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) = R_t$ and an error message otherwise.

Show that for every matrix $A$ of $\mathrm{O}(2)$ such that $\det(A) = -1$, there exists a real $t$ such that $$A = \left(\begin{array}{cr} \cos(2t) & \sin(2t) \\ \sin(2t) & -\cos(2t) \end{array}\right)$$

Let $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ where $V ^ { \prime }$ is a $\mathbb { K }$-vector space of finite dimension. Prove that $O ( q )$ is a subgroup of $\mathrm { GL } ( V )$ and that if $q \cong q ^ { \prime }$, then $O ( q )$ and $O \left( q ^ { \prime } \right)$ are two isomorphic groups.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension, $q \in \mathcal { Q } ( V )$ and $v , w \in V$ be two distinct vectors of $V$ such that $q ( v ) = q ( w ) \neq 0$.
We want to show in this question that there then exists an isometry $h \in O ( q )$ such that $h ( v ) = w$.
(a) Let $x \in V$ such that $q ( x ) \neq 0$. We denote by $s _ { x }$ the endomorphism of $V$ defined by $y \mapsto s _ { x } ( y ) = y - 2 \frac { \widetilde { q } ( x , y ) } { q ( x ) } x$. Show that $s _ { x }$ and $- s _ { x }$ belong to $O ( q )$.
(b) Suppose here that $q ( w - v ) \neq 0$. Show that the map $s _ { w - v }$ is an isometry such that $s _ { w - v } ( v ) = w$.
(c) Suppose here that $q ( w - v ) = 0$. Show that there exists an isometry $g \in O ( q )$ such that $g ( v ) = w$ and conclude.

We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ Let $L \in \mathcal { M } _ { p + 1 } ( \mathbb { R } )$ and $f$ the endomorphism of $\mathbb { R } ^ { p + 1 }$ canonically associated.
Show that the following three assertions are equivalent:
i. $L \in O ( 1 , p )$;
ii. $\forall \left( v , v ^ { \prime } \right) \in \left( \mathbb { R } ^ { p + 1 } \right) ^ { 2 } , \varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$;
iii. $\forall v \in \mathbb { R } ^ { p + 1 } , q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v )$.

We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ If $L = \left( l _ { i , j } \right) _ { i , j } \in O ( 1 , p ) , v = ( 1,0 , \ldots , 0 )$ and $v ^ { \prime } = ( 0,1,0 , \ldots , 0 )$, give the equations on the $l _ { i , j }$ corresponding to $$\varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right) , \quad q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v ) \quad \text { and } \quad q _ { p + 1 } \left( f \left( v ^ { \prime } \right) \right) = q _ { p + 1 } \left( v ^ { \prime } \right)$$ What do we obtain similarly with ${ } ^ { t } L$ ?

Let $a , b , c$ and $d$ be four real numbers. We consider the matrix of $\mathcal { M } _ { 2 } ( \mathbb { R } )$ $$L = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$$ Write the equations on $a , b , c , d$ expressing the membership of $L$ in $O ( 1,1 )$.

Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ( 1,3 )$. Show the inequality $\ell _ { 1,1 } ^ { 2 } \geqslant 1$.

In the usual Euclidean space $\mathbb { R } ^ { 3 }$, show that, for all vectors $u$ and $v$ of $\mathbb { R } ^ { 3 }$ of the same norm, there exists a rotation $r$ such that $r ( u ) = v$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ We recall that, if $M$ is a matrix in $\mathcal{M}_n(\mathbf{R})$, we say that $M$ is tangent to $G$ at $I_n$ if there exist $\varepsilon > 0$ and an application $\left.\gamma : \right]-\varepsilon, \varepsilon[ \rightarrow G$, differentiable, such that $\gamma(0) = I_n$ and $\gamma'(0) = M$. The set of matrices tangent to $G$ at $I_n$ is called the tangent space to $G$ at $I_n$, and is denoted $\mathcal{T}_{I_n}(G)$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{14}$ ▷ Prove the inclusion $\mathcal{A}_G \subset \mathcal{T}_{I_n}(G)$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ The tangent space to $G$ at $I_n$ is denoted $\mathcal{T}_{I_n}(G)$.
$\mathbf{17}$ ▷ Show that, in the particular cases $G = \mathrm{SL}_n(\mathbf{R})$ and $G = \mathrm{O}_n(\mathbf{R})$, we have $\mathcal{T}_{I_n}(G) = \mathcal{A}_G$.

Let $F$ be a vector subspace of a symplectic space $(E,\omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega(x,y) = 0 \}$. Show that the restriction $\omega _ { F }$ of $\omega$ to $F ^ { 2 }$ defines a symplectic form on $F$ if and only if $F \oplus F ^ { \omega } = E$.

The set of real symplectic matrices is defined as $$\mathrm { Sp } _ { n } ( \mathbb { R } ) = \mathrm { Sp } _ { 2 m } ( \mathbb { R } ) = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid M ^ { \top } J M = J \right\}$$ where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is a subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$, stable under transposition and containing the matrix $J$.

We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ the set of real symplectic and orthogonal matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$. We equip $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space of finite dimension. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } )$ is a compact subgroup of the symplectic group $\operatorname { Sp } _ { n } ( \mathbb { R } )$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and let $\lambda$ and $\mu$ be real numbers. The symplectic transvections are defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\tau _ { a } ^ { \mu } \circ \tau _ { a } ^ { \lambda } = \tau _ { a } ^ { \lambda + \mu }$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\operatorname { det } \left( \tau _ { a } ^ { \lambda } \right) > 0$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Is the inverse $\left( \tau _ { a } ^ { \lambda } \right) ^ { - 1 }$ still a symplectic transvection?

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) \neq 0$. Show that there exists $\lambda \in \mathbb { R }$ such that $\tau _ { y - x } ^ { \lambda } ( x ) = y$, where $\tau_a^{\lambda}(x) = x + \lambda \omega(a,x)a$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) = 0$. Show that there exists a vector $z \in E$ such that $\omega ( x , z ) \neq 0$ and $\omega ( y , z ) \neq 0$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Prove the following lemma: For all non-zero vectors $x$ and $y$ of $E$, there exists a composition $\gamma$ of at most two symplectic transvections of $E$ such that $\gamma ( x ) = y$.