Given's rotation easy understanding problem

Question

Consider $ \begin{pmatrix} 0 & 5 \\ 2 & 3 \end{pmatrix} $

Give a Given's rotation, which maps the first line(row) of this matrix on a multiple of $\begin{pmatrix} 1 \\ 0 \end{pmatrix} $
I do not really understand this formulation. Does it mean to eliminate the first element in the second line($0$ instead of $2$)?

Answer 1

Let $c= \cos \theta$ and $s = \sin \theta$, find a matrix $\begin{bmatrix} c & s \\ - s & c\end{bmatrix}$ such that

$$\begin{bmatrix} c & s \\ - s & c\end{bmatrix} \begin{bmatrix} 0 \\ 2\end{bmatrix} = 2\begin{bmatrix}1 \\ 0 \end{bmatrix}$$

The $2$ is due to rotation doesn't change the length.

Alternatively, $$\begin{bmatrix} c & s \\ - s & c\end{bmatrix} \begin{bmatrix} 0 \\ 2\end{bmatrix} = -2\begin{bmatrix}1 \\ 0 \end{bmatrix}$$