Prove the columns of the rotation matrix Rθ form a linearly independent set the question is as above, I don't know how the rotation matrix works in this question. After giving a formula of proving linear independence, I have no more ideas.

$R_\theta = \left( \begin{array}{cc} \cos (\theta) & -\sin (\theta) \\\ \sin (\theta) & \cos (\theta) \\\ \end{array} \right)$

To be linearly _de_ pendent, we must be able to express one row as a (real) multiple $\alpha$ of the other row. That would mean:

* $\alpha \cos \theta = \sin \theta$
* $-\alpha \sin \theta = \cos \theta$



This implies that $\alpha = \sqrt{-1}$, and hence is not real. Hence these rows are not _de_ pendent, and hence are _independent_.