It is an exercise in Artin's Algebra that "describe geometrically composition of two rotations (along axis through $0$) in $\mathbb{R}^3$".
I thought in the following linear algebra way: if $l,m$ are axis of rotations (passing through $0$) then we can think of $l$, $m$ as one dimensional subspaces. Since rotations stabilize $l$ and $m$ respectively, it follows that they stabilize $l^{\perp}$ and $m^{\perp}$, which are two-dimensional subspaces, so their intersection is $1$-dimensional, say a line $n$ passing through $0$.
Then composition of rotations along $l$ and $m$ is rotation along line $n$.
Is this the correct way? Otherwise, give me a hint.
Best Answer
It's not correct since a rotation with one subspace preserved followed by another rotation with another subspace preserved does not result in the intersection being preserved. I'm not sure what is meant by "geometrically describe". We know that the only norm-preserving transformations are translations, rotations and reflections, and since norm-preserving transformations are closed under composition, we know that combining two rotations must be another norm-preserving transformation. Also we know that rotations preserve an orthogonal basis and its orientation (determinant), so combining them also does. In particular, we know that rotations are essentially orthogonal matrices with determinant one, so a combination is also. To find the axis of rotation is equivalent to finding an eigenvector.