For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Verify that, for all $q \in \mathbb{H}$, $q^*$ is the transpose of the matrix whose coefficients are the conjugates of the coefficients of $q$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $(qr)^* = r^* q^*$. c) Show that $q^{**} = q$ for all $q \in \mathbb{H}$ and that $q \mapsto q^*$ is an automorphism of the $\mathbb{R}$-vector space $\mathbb{H}$. d) For $q \in \mathbb{H}$, express $qq^*$ in terms of $N(q)$. Deduce the relation valid for all $(q, r) \in \mathbb{H}^2$ $$N(qr) = N(q)N(r)$$
For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$.\\
a) Verify that, for all $q \in \mathbb{H}$, $q^*$ is the transpose of the matrix whose coefficients are the conjugates of the coefficients of $q$.\\
b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $(qr)^* = r^* q^*$.\\
c) Show that $q^{**} = q$ for all $q \in \mathbb{H}$ and that $q \mapsto q^*$ is an automorphism of the $\mathbb{R}$-vector space $\mathbb{H}$.\\
d) For $q \in \mathbb{H}$, express $qq^*$ in terms of $N(q)$. Deduce the relation valid for all $(q, r) \in \mathbb{H}^2$
$$N(qr) = N(q)N(r)$$