\"{U}ber die Winkel zwischen Unterr\"{a}umen

We prove a metric statement about approximation of a $n$-dimensional linear subspace $A$ in $\mathbb{R}^d$ by $n$-dimensional rational subspaces. We consider the problem of finding a rational subspace $B$ of bounded height $H=H(B)$ for which the angle of inclination $\psi (A,B) $ is small in terms of $H$. In the simplest case $d=4, n=2$ we give a partial solution of a problem formulated by W.M. Schmidt in 1967.