Let $\mathrm { O } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of orthogonal matrices, i.e., matrices $A$ satisfying $A ^ { \operatorname { tr } } A = I$. Let $\mathrm { B } _ { + } ( 2 , \mathbb { R } )$ be the subgroup of $\mathrm { GL } ( 2 , \mathbb { R } )$ consisting of upper triangular matrices with positive entries on the diagonal.
(A) Let $A \in \mathrm { GL } ( 2 , \mathbb { R } )$. Show that there exist $A _ { o } \in \mathrm { O } ( 2 , \mathbb { R } )$ and $A _ { b } \in \mathrm { B } _ { + } ( 2 , \mathbb { R } )$ such that $A = A _ { o } A _ { b }$. (Hint: use appropriate elementary column operations.)
(B) Show that the map $$\phi : \mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } ) \longrightarrow \mathrm { GL } ( 2 , \mathbb { R } ) \quad \left( A ^ { \prime } , A ^ { \prime \prime } \right) \mapsto A ^ { \prime } A ^ { \prime \prime }$$ is injective.
(C) Show that $\mathrm { GL } ( 2 , \mathbb { R } )$ is homeomorphic to $\mathrm { O } ( 2 , \mathbb { R } ) \times \mathrm { B } _ { + } ( 2 , \mathbb { R } )$. (Hint: first show that the map $A \mapsto A _ { b }$ is continuous.)

Let $F$ be a field of characteristic $p > 0$ and $V$ a finite-dimensional $F$-vector-space. Let $\phi \in \mathrm { GL } ( V )$ be an element of order $p ^ { 3 }$. Show that there exists a basis of $V$ with respect to which $\phi$ is given by an upper-triangular matrix with 1's on the diagonal.

Let $U ( n )$ be the group of $n \times n$ unitary complex matrices. Let $P \subset U ( n )$ be the set of all finite order elements of $U ( n )$, that is, $P = \left\{ X \in U ( n ) \mid X ^ { m } = 1 \text{ for some } m \geq 1 \right\}$. Show that $P$ is dense in $U ( n )$.

Let $X \subseteq \mathbb { R } ^ { n }$ and $p \in X$. By a tangent vector of $X$ at $p$, we mean $\gamma ^ { \prime } ( 0 )$, where $\gamma : ( - \epsilon , \epsilon ) \longrightarrow X$ is a differentiable function with $\gamma ( 0 ) = p$. ($\epsilon \in \mathbb { R } , \epsilon > 0$.) The tangent space of $X$ at $p$ is the $\mathbb { R }$-vector space of all the tangent vectors at $p$. Think of $\mathrm { GL } _ { n } ( \mathbb { C } )$ as a subspace of $\mathbb { R } ^ { 2 n ^ { 2 } }$, with the euclidean topology.
Let $G : = \left\{ A \in \mathrm { GL } _ { 2 } ( \mathbb { C } ) \mid A ^ { * } A = A A ^ { * } = I _ { 2 } , \operatorname { det } A = 1 \right\}$.
(A) (2 marks) Show that every tangent vector of $\mathrm { GL } _ { n } ( \mathbb { C } )$ at $I _ { n }$ is of the form $\gamma _ { A } ^ { \prime } ( 0 )$ where $A$ is a $n \times n$ complex matrix and $\gamma _ { A } : \mathbb { R } \longrightarrow \mathrm { GL } _ { n } ( \mathbb { C } )$ is the function $t \mapsto e ^ { t A }$.
(B) (3 marks) Show that the tangent space of $G$ at $I _ { 2 }$ is $V : = \left\{ \left. \left[ \begin{array} { c c } i a & z \\ - \bar { z } & - i a \end{array} \right] \right\rvert \, a \in \mathbb { R } , z \in \mathbb { C } \right\}$.
(C) (5 marks) Consider the homeomorphism $\Phi : G \longrightarrow \mathbb { S } ^ { 3 }$ (where $\mathbb { S } ^ { 3 }$ denotes the unit sphere in $\mathbb { R } ^ { 4 }$) given by $$\left[ \begin{array} { c c } \alpha & \beta \\ \bar { \beta } & \bar { \alpha } \end{array} \right] \mapsto ( \Re ( \alpha ) , \Im ( \alpha ) , \Re ( \beta ) , \Im ( \beta ) )$$ Define a 'multiplication' on $V$ by $[ A , B ] = \frac { A B - B A } { 2 }$. Determine the multiplication on the tangent space at $\Phi \left( I _ { 2 } \right)$ induced by the derivative $D \Phi$. (Hint: The map $( A , B ) \longrightarrow [ A , B ]$ is $\mathbb { R }$-bilinear.)

Let $f$ be an antisymmetric endomorphism of $E$. What is $f^{2} = f \circ f$ if $f$ is an orthogonal automorphism and antisymmetric of $E$?

In this question only, we assume $n = 2$. Explicitly give a vector space of dimension 2, formed of similarity matrices. Deduce from this, carefully, that $d_{2} = 2$.

In this question only, we assume $n$ is odd. If $f, g$ belong to $GL(E)$, show that there exists $\lambda \in \mathbb{R}$ such that $f + \lambda g$ is non-invertible. One may reason by considering the characteristic polynomial of $fg^{-1}$. Deduce that $d_{n} = 1$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that for all $i \in \{1, 2, \ldots, d-1\}$, $f_{i}^{*} + f_{i}$ is collinear to $\operatorname{Id}_{E}$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that there exists a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ such that for all $i \in \{1, 2, \ldots, d-1\}$, $g_{i}$ is antisymmetric (one will seek $g_{i}$ as a combination of $f_{i}$ and $\operatorname{Id}_{E}$).

Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. We fix a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ with for all $i$, $g_{i}$ antisymmetric. a) Show that for all $i \neq j$, $g_{i}g_{j} + g_{j}g_{i}$ is collinear to $\operatorname{Id}_{E}$. b) Show that we define a scalar product on $\mathscr{L}(E)$ by setting, for all $f, g$ of $\mathscr{L}(E)$: $(f \mid g) = \operatorname{tr}(f^{*}g)$. We consider, in the rest of this question, a basis $(h_{1}, \ldots, h_{d-1})$ of $\operatorname{Vect}(g_{1}, \ldots, g_{d-1})$ orthogonal for this scalar product. c) Show that the $h_{i}$ are antisymmetric and satisfy: $\forall i \neq j, h_{i}h_{j} + h_{j}h_{i} = 0$. What should be done so that the $h_{i}$ are also orthogonal automorphisms?

Conversely, let $(h_{1}, \ldots, h_{d-1})$ be a family of $\mathscr{L}(E)$ such that the $h_{i}$ are antisymmetric orthogonal automorphisms satisfying for all $i \neq j$, $h_{i}h_{j} + h_{j}h_{i} = 0$. Show that $\operatorname{Vect}\left(\operatorname{Id}_{E}, h_{1}, \ldots, h_{d-1}\right)$ is a vector subspace of $\mathscr{L}(E)$, of dimension $d$, included in $\operatorname{Sim}(E)$.

Let $p$ be an odd integer such that $\operatorname{dim}(E) = n = 2p$. We assume that there exist $d \geqslant 3$ and a family $(f_{1}, f_{2}, \ldots, f_{d-1})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let $x \in E$ of norm 1. a) Show that $(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is an orthonormal family, and that $S = \operatorname{Vect}(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is stable under $f_{1}$ and $f_{2}$. b) Deduce that $d_{n-4} \geqslant 3$.

In this question, $n = 2p$, with $p$ an odd integer. Show that $d_{n} = 2$.

In this section, the dimension of $E$ is 4. We assume that there exists a vector subspace of $\mathscr{L}(E)$ of dimension 4 included in $\operatorname{Sim}(E)$. We then consider, in accordance with I.D.4, a family $(f_{1}, f_{2}, f_{3})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let a fixed vector $x \in E$ of norm 1. a) Justify that the family $B = (x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is a basis of $E$ then show that there exist real numbers $\alpha, \beta, \gamma, \delta$ such that: $$f_{3}(x) = \alpha x + \beta f_{1}(x) + \gamma f_{2}(x) + \delta f_{1}f_{2}(x)$$ Show that $\alpha = \beta = \gamma = 0$ and that $\delta \in \{-1, 1\}$. b) Show that $f_{3} = \delta f_{1}f_{2}$. If necessary, by replacing $f_{3}$ with its opposite, we assume in what follows that $f_{3} = f_{1}f_{2}$. c) If $x_{0}, x_{1}, x_{2}, x_{3}$ are real numbers, give the matrix $M(x_{0}, x_{1}, x_{2}, x_{3})$ in $B$ of the endomorphism $x_{0}\operatorname{Id}_{E} + x_{1}f_{1} + x_{2}f_{2} + x_{3}f_{3}$.

In this section, the dimension of $E$ is 4. Verify that for all $(x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4}$, $M(x_{0}, x_{1}, x_{2}, x_{3})$ is a similarity matrix. What can we conclude?

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. By using $f_{4}$, show that $f_{3}$ cannot be equal to $\pm f_{1}f_{2}$.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Show that $f_{1}f_{2}f_{3}$ is an orthogonal automorphism, symmetric and not collinear to $\operatorname{Id}_{E}$.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. What is the spectrum of $f_{1}f_{2}f_{3}$? Show that there exists $x \in E$ of norm 1 such that $\langle f_{1}f_{2}f_{3}(x), x \rangle = 0$. We fix such an $x$ for the rest.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. We fix $x \in E$ of norm 1 such that $\langle f_{1}f_{2}f_{3}(x), x \rangle = 0$. Show that $F = (x, f_{1}(x), f_{2}(x), f_{3}(x), f_{1}f_{2}(x), f_{1}f_{3}(x), f_{2}f_{3}(x), f_{1}f_{2}f_{3}(x))$ is an orthonormal family.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. We set $V = \operatorname{Vect}(F)$ where $F = (x, f_{1}(x), f_{2}(x), f_{3}(x), f_{1}f_{2}(x), f_{1}f_{3}(x), f_{2}f_{3}(x), f_{1}f_{2}f_{3}(x))$. It is therefore a vector subspace of $E$ of dimension 8. a) Show that $V^{\perp}$ is stable under $f_{1}, f_{2}, f_{3}$. b) We denote by $f_{i}^{\prime}$ the endomorphism induced by $f_{i}$ on $V^{\perp}$, $i = 1, 2, 3$. Justify that there exists $\delta^{\prime} \in \{-1, 1\}$ such that $f_{3}^{\prime} = \delta^{\prime} f_{1}^{\prime} f_{2}^{\prime}$. If necessary, by replacing $f_{3}$ by $-f_{3}$, we consider for the rest that $f_{3}^{\prime} = f_{1}^{\prime} f_{2}^{\prime}$. c) Let $e$ be fixed in $V^{\perp}$, of norm 1. By proceeding as in II.B.1.a) (but this is not to be redone), one can show that $(e, f_{1}(e), f_{2}(e), f_{1}f_{2}(e))$ is an orthonormal basis of $V^{\perp}$. By noting that $f_{3}(e) = f_{1}f_{2}(e)$, use this basis to show that: $\forall y \in V^{\perp}, f_{4}(y) \in V$. Thus $W = f_{4}(V^{\perp})$ is a vector subspace of $V$ of dimension 4. d) Show that the sum of $W$ and $V^{\perp}$ is direct and that $W \oplus V^{\perp}$ is stable under $f_{1}, f_{2}, f_{3}, f_{4}$. Then reach a contradiction.

In this section, the dimension of $E$ is 12. Deduce the value of $d_{12}$.

In this section, the dimension of $E$ is 8. Show that, for all $(x_{0}, \ldots, x_{7}) \in \mathbb{R}^{8}$, $$\left(\begin{array}{cccccccc} x_{0} & -x_{1} & -x_{2} & -x_{4} & -x_{3} & -x_{5} & -x_{6} & -x_{7} \\ x_{1} & x_{0} & -x_{4} & x_{2} & -x_{5} & x_{3} & -x_{7} & x_{6} \\ x_{2} & x_{4} & x_{0} & -x_{1} & -x_{6} & x_{7} & x_{3} & -x_{5} \\ x_{4} & -x_{2} & x_{1} & x_{0} & x_{7} & x_{6} & -x_{5} & -x_{3} \\ x_{3} & x_{5} & x_{6} & -x_{7} & x_{0} & -x_{1} & -x_{2} & x_{4} \\ x_{5} & -x_{3} & -x_{7} & -x_{6} & x_{1} & x_{0} & x_{4} & x_{2} \\ x_{6} & x_{7} & -x_{3} & x_{5} & x_{2} & -x_{4} & x_{0} & -x_{1} \\ x_{7} & -x_{6} & x_{5} & x_{3} & -x_{4} & -x_{2} & x_{1} & x_{0} \end{array}\right)$$ is a similarity matrix. What can we deduce from this?

Conjecture the value of $d_{n}$ in the general case.

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).

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 $k_{/F} = f$.