Define a rotation of $V$ to be a real unitary map $A$ of $V$ whose determinant is 1. Show that the matrix of $A$ relative to an orthogonal basis of $V$ is of type
$\begin{bmatrix}a&-b\\b&a\end{bmatrix}$
for some real numbers $a,b$ such that $a^2+b^2=1$.
SOLUTION. Let $\{v_1,v_2\}$ be an orthogonal basis for $V$. Let $w_i=Av_i$ and
$w_1=av_1+bv_2$
$w_2=cv_1+dv_2$
The matrix representing $V$ in the chosen basis is
$\begin{bmatrix}a&c\\b&d\end{bmatrix}$.
Then, since $\langle Av_i,Av_i\rangle=\langle v_i,v_i\rangle$ we have
$(a^2-1)\langle v_1,v_1\rangle + b^2\langle v_2,v_2\rangle=0$
$(c^2)\langle v_1,v_1\rangle + (d^2-1)\langle v_2,v_2\rangle=0$
But $dw_1-bw_2=(ad-bc)v_1=v_1$,so
$\langle v_1,v_1\rangle=\langle A(dv_1-dv_2),A(dv_1-dv_2)\rangle=d^2\langle v_1,v_1\rangle + b^2\langle v_2,v_2\rangle$,
thus implies $a^2=d^2$ and $b^2=c^2$. Moreover,
$0=\langle v_1,v_2\rangle=\langle Av_1,Av_2\rangle=ac\langle v_1,v_1\rangle+bd\langle v_2,v_2\rangle$,
so $ac$ and $bd$ are of opposite signs and therefore the matrix $A$ has the desired form.Solutions Manual for Lang´s Linear Algebra, Rami Sharcharchi
Questions:
1) Since the basis are assumed orthogonal and not orthonormal. How can the author possible derive $(a^2-1)\langle v_1,v_1\rangle + b^2\langle v_2,v_2\rangle=0$? How does he knows the last equality? What is the intuition?
2) $dw_1-bw_2=(ad-bc)v_1=v_1$ How does the author know $(ad-cb)=1$ if we are not working with orthonormal basis?
Thanks in advance!