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 return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $\left( V _ { i } \right) _ { 1 \leq i \leq 3 }$ be three $\mathbb { K }$-vector spaces of finite dimension and $q _ { i } \in \mathcal { Q } \left( V _ { i } \right)$ for $1 \leq i \leq 3$ satisfying $q _ { 1 } \perp q _ { 3 } \cong q _ { 2 } \perp q _ { 3 }$. Show that $q _ { 1 } \cong q _ { 2 }$.
Hint: one may reason by induction and use questions 17 and 18.

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 and $q \in \mathcal { Q } ( V )$. Show that there exists a unique non-negative integer $m$ and an anisotropic quadratic form $q _ { \text {an} }$, unique up to isometry, such that $q \cong q _ { an } \perp m \cdot h$ where $m \cdot h = h \perp \cdots \perp h$ is the orthogonal sum of $m$ copies of $h$ and $h$ is the quadratic form defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (introduced in question 6b).
Hint: one may use question 6b and the previous question.

Determine a pair $(A, \vec{b})$ in $\mathrm{SO}(2) \times \mathbb{R}^2$ such that $M(A, \vec{b}) = I_3$.

Let $(A, \vec{b})$ and $(A^{\prime}, \vec{b}^{\prime})$ be in $\mathrm{SO}(2) \times \mathbb{R}^2$. Show that $M(A, \vec{b}) M\left(A^{\prime}, \vec{b}^{\prime}\right) = M\left(A A^{\prime}, A \vec{b}^{\prime} + \vec{b}\right)$.

Show that the elements of $G$ are invertible and explicitly determine the inverse of $M(A, \vec{b})$.

Prove that $G$ is a subgroup of $\mathrm{GL}_3(\mathbb{R})$.

Is the application $\Phi : \left\{ \begin{array}{cll} G & \rightarrow & \mathbb{R}^2 \\ M(A, \vec{b}) & \mapsto & \vec{b} \end{array} \right.$ surjective? Is it injective?

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Determine $\Psi\left(M\left(I_2, \overrightarrow{0}\right)\right)$.

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Verify that $\Psi\left(M\left(R_\theta, q\vec{u}_\theta\right)\right) = \Delta\left(q, \vec{u}_\theta\right)$; deduce that $\Psi$ is surjective.

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Let $H$ be the set of matrices $M(A, \vec{b})$ of $G$ such that $\Psi(M(A, \vec{b})) = \Delta\left(0, \vec{e}_1\right)$.
a) Describe the elements of $H$.
b) Show that $H$ is a subgroup of $G$.
c) Show that for all $g$ in $G$ and all $h$ in $H$, we have $\Psi(gh) = \Psi(g)$.

If $f$ is a function defined on $\mathbb{R}^2$, we denote by $f^*$ the function $f \circ \Phi$, defined on $G$ by $f^*(g) = f(\Phi(g))$ where $\Phi : G \rightarrow \mathbb{R}^2$ is the function introduced in question I.A.5.
Prove that for all $g$ in $G$ and $r$ such that $\Phi(r) = \overrightarrow{0}$ we have $f^*(gr) = f^*(g)$.

We now assume that $f$ satisfies the hypotheses allowing us to define its Radon transform.
Demonstrate that if two lines $\Delta\left(q_1, \vec{u}_{\theta_1}\right)$ and $\Delta\left(q_2, \vec{u}_{\theta_2}\right)$ coincide, then $\hat{f}\left(q_1, \theta_1\right) = \hat{f}\left(q_2, \theta_2\right)$.

We define $\hat{f}^\star$ on $G$ by composing $\hat{f}$ with $\Psi$: we set, for all $g \in G$, $\hat{f}^\star(g) = \hat{f}(\Psi(g))$.
Demonstrate that $\hat{f}^\star$ is $H$-invariant, that is, for all $g \in G$ and $h \in H$, $\hat{f}^\star(gh) = \hat{f}^\star(g)$.

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that $\mathcal{S}$ is a vector space over $\mathbb{R}$.

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $P$ is a polynomial function and if $f$ is in $\mathcal{S}$, then $Pf$ belongs to $\mathcal{S}$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Justify that $T'$ is a distribution on $\mathcal{D}$.

Let $m \in \mathbb{N}$. Justify that the vector space $\mathcal{P}_m$ is finite-dimensional and determine its dimension.

Determine a harmonic polynomial of degree 1, then of degree 2.

a) Show that the set of harmonic polynomials is a vector subspace of $\mathcal{P}$.
b) For all $m \geqslant 2$, we denote by $\Delta_m$ the restriction of $\Delta$ to $\mathcal{P}_m$. Show that $\operatorname{dim}(\operatorname{ker} \Delta_m) \geqslant 2m+1$.
c) What can be deduced about the dimension of the vector space of harmonic polynomials?

Determine a harmonic polynomial $H$ that satisfies $H(x,y) = f(x,y)$ for all $(x,y) \in C(0,1)$, where $f(x,y) = xy$.

Determine a harmonic polynomial $H$ that satisfies $H(x,y) = f(x,y)$ for all $(x,y) \in C(0,1)$, where $f(x,y) = x^4 - y^4$.

Let $f : \Omega \rightarrow \mathbb{R}$ be a harmonic application of class $C^2$ such that $\partial_1 f$ and $\partial_2 f$ are of class $C^2$ on $\Omega$. Show that the applications $\partial_1 f$ and $\partial_2 f$ are also harmonic on $\Omega$.

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ By which geometric transformation(s) is the set $\Omega_{x_0, y_0, \lambda}$ the image of $\Omega$? Justify that $\Omega_{x_0, y_0, \lambda}$ is an open set of $\mathbb{R}^2$.