Find the linear dependence using the rotation matrix

Question

My attempt: The only thing that I know is rotating $[c_1 , c_2 ]^T$ by an angle $\theta$ in the counter-clockwise direction is the same as multiplying the rotation matrix by $[c_1 , c_2 ]^T$. How to use this to find the linear dependence between $y$ and $x$?

Answer 1

As you say, all you have done is multiply by a rotation matrix of angle $\theta$ in the two dimensional subspace spanned by $u$ and $v$. If we extend the orthonormal basis $(u,v)$ of this subspace to a (if you want, orthonormal) basis of $\mathbb{R}^m$, then $Q$ does not affect the other basis vectors, and we may write the matrix as $$ Q = \begin{bmatrix} R_\theta & 0 \\ 0 & I_{m-2 \times m-2} \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta & 0 & \dots & 0 \\ \sin \theta & \cos \theta & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ \end{bmatrix} $$ in the basis where the first two vectors are $(u,v)$.