Let $\text{SO}_n$ denote the special orthogonal group and $G(n,k)$ be the set of all $k$-dimensional linear subspaces of $\mathbb{R}^n$. Show that $f:\text{SO}_n\to G(n,k)$ given by $f(O)=OL$, where $O\in \text{SO}_n$ and $L\in G(n,k)$ ($L$ is fixed) is surjective.
I'm not really sure how to show this and could use some help.
You want to show that you can rotate any $k$-dimensional subspace to any other. Pick a basis $B$ of subspace $L_1,$ complete it to a basis $\mathbf{B}$ of $\mathbb{R}^n.$ Now pick a basis $C$ of $L_2,$ complete it to a basis $\mathbf{C}$ of $\mathbb{R}^n.$ Now, find an orthogonal transformation sending $\mathbf{B}$ to $\mathbf{C}.$