I am attempting to prove that three axes of rotation are linearly independent and so form a basis for $\mathbb{R}^3$. I assert that given the fact that rotations about each axis involve no rotation about the other axes proves that they are linearly independent and so form a set spanning $\mathbb{R}^3$.
With that being said, I am unfamiliar with linear independence in this context, so I am unsure as to whether this is a valid strategy.
$Thanks$
A (pairwise) orthogonal set is always linearly independent: Assume $v_1, v_2, v_3$ pairwise orthogonal and wlog , orthonormal, and assume $c_1v_1+c_2v_2+ c_3v_3=0$. Then take, e.g., $v_3$ and use: $\langle v_3, c_1v_1+c_2v_2+c_3v_3\rangle =0= c_3$. Do similar for $v_1,v_2$