# More on rotation matrices: a basis for $SO(3)$?

Consider the group $SO(3)$. The rotation around the $x$-axis is represented by the matrix

$$R_x = \left ( \begin{array}{ ccc } 1 & 0 & 0 \\ 0 & \cos \Theta & - \sin \Theta \\ 0 & \sin \Theta & \cos \Theta \end{array}\right )$$

Similarly, for the rotations around the other axes. It is very easy to verify that these three are linearly independent.

I wanted to show that they form a basis for $SO(3)$. But this is where I got stuck. The idea should be to consider an arbitrary rotation and then show that it is a linear combination of these $3$ matrices. But I don't know what a generic rotation matrix looks like. So maybe there is a different way of showing this or maybe it is even false.

• $SO(3)$ is not a group under addition, so you're not going to get very far with showing something is a basis in the sense of linear algebra. The relevant notion is of a generating set of a group. Nov 9 '14 at 1:36
• Consider the action of $SO(3)$ on $S^2$. Nov 9 '14 at 1:55
Basis is the wrong word. However, you are correct; an oriented (orthogonal) basis can be rotated to the standard $x,y,z$ directions in no more than 3 standard rotations. One to put the $x$ vector in the $xz$ plane. a second rotation to send it to the positive $x$ axis. One final rotation in the $yz$ plane to take those vectors to their proper positions.