Paper 1, Section II, A

(a) Explain what is meant by saying that a $2 \times 2$ real transformation matrix

\begin{aligned} &A=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \text { preserves the scalar product with respect to the Euclidean metric } \\ &I=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) \text { on } \mathbb{R}^{2} . \end{aligned}

Derive a description of all such matrices that uses a single real parameter together with choices of $\operatorname{sign}(\pm 1)$. Show that these matrices form a group.

(b) Explain what is meant by saying that a $2 \times 2$ real transformation matrix $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ preserves the scalar product with respect to the Minkowski metric $J=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$ on $\mathbb{R}^{2}$

Consider now the set of such matrices with $a>0$. Derive a description of all matrices in this set that uses a single real parameter together with choices of sign $(\pm 1)$. Show that these matrices form a group.

(c) What is the intersection of these two groups?