Let $f$ be an antisymmetric endomorphism of $E$. Show that: $\forall x \in E, \langle x, f(x) \rangle = 0$.

Let $f$ be an antisymmetric endomorphism of $E$. Let $g$ be an antisymmetric endomorphism of $E$, such that $fg = -gf$. Show that: $\forall x \in E, \langle f(x), g(x) \rangle = 0$.

Prove that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ is indeed a quadratic form on $\mathbb { K } ^ { n }$, where $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ denotes the quadratic form $q$ defined on $\mathbb { K } ^ { n }$ by the formula $$q \left( x _ { 1 } , \ldots , x _ { n } \right) = a _ { 1 } x _ { 1 } ^ { 2 } + \cdots + a _ { n } x _ { n } ^ { 2 }$$

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$
(a) Show that we define a group law $*$ on $\mathbf{L}$ by setting for $M, N \in \mathbf{L}$: $$M * N = M + N + \frac{1}{2}[M, N]$$ Explicitly determine the inverse of $M_{p,q,r}$.
(b) Determine the matrices $M_{p,q,r} \in \mathbf{L}$ that commute with all elements of $\mathbf{L}$ for the law $*$. Is $(\mathbf{L}, *)$ commutative?

Verify that $\delta$ is a neutral element for the operation $*$.

Deduce that $*$ is commutative.

Similarly, by exploiting the set $\mathcal{C}_n^{\prime} = \left\{ \left( d_1, d_2, d_3 \right) \in \left( \mathbb{N}^* \right)^3 \mid d_1 d_2 d_3 = n \right\}$, show that $*$ is associative.

Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We denote by $G L _ { r } ( A )$ the set of matrices of $M _ { r } ( A )$ that satisfy the equivalent properties of question III.3.d). a) Show that matrix multiplication induces a group structure on $G L _ { r } ( A )$. b) We define a relation on $M _ { s , r } ( A )$ by $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$. Show that this is an equivalence relation.

For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.

• [(a)] Verify that for all $a, b \in \mathbb{R}^{d}$ and $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $|\phi_{g}(a) - \phi_{g}(b)| = |a - b|$.
• [(b)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $\phi_{g} = \phi_{g^{\prime}}$ if and only if $g = g^{\prime}$.
• [(c)] Show that there exists a unique $e \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{e}$ is the identity map on $\mathbb{R}^{d}$, that is $\phi_{e}(x) = x$ for all $x \in \mathbb{R}^{d}$.

For which values of $d$ do we have $gg^{\prime} = g^{\prime}g$ for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$?

Binary operations $*$, $\oplus$, $\odot$ defined on the set of rational numbers
I. $a * b = a - b$ II. $a \oplus b = a + b + ab$ III. $a \odot b = \frac{a+b}{5}$
are defined as follows. Accordingly, which of these operations satisfy the associative property?
A) Only I
B) Only II
C) Only III
D) I and II
E) II and III

The $\Delta$ operation on the set $\mathrm { A } = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } \}$ is defined by the table below. For example, a $\Delta \mathrm { d } = \mathrm { c }$ and $\mathrm { d } \Delta \mathrm { a } = \mathrm { a }$.

$\Delta$	a	b	c	d	e
a	a	b	a	c	d
b	c	b	b	a	e
c	a	b	c	d	e
d	a	a	d	d	b
e	e	e	e	d	a

According to this table, which of the following subsets of set A

• $\mathrm { K } = \{ \mathrm { b } , \mathrm { c } , \mathrm { d } \}$
• $\mathrm { L } = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } \}$
• $\mathrm { M } = \{ \mathrm { c } , \mathrm { d } , \mathrm { e } \}$

are closed under the $\Delta$ operation?
A) Only K
B) Only L
C) K and L
D) K and M
E) L and M