- #1
jostpuur
- 2,116
- 19
If [itex]N>2[/itex] and [itex]A\in\textrm{SO}(N)[/itex] are arbitrary, does there exist subspaces [itex]V_1,V_2\subset\mathbb{R}^N[/itex] such that
[tex]
V_1+ V_2 = \mathbb{R}^N,\quad\quad \textrm{dim}(V_1)=2,\quad \textrm{dim}(V_2)=N-2
[/tex]
and such that the restriction of [itex]A[/itex] to [itex]V_1[/itex] belongs to [itex]\textrm{SO}(2)[/itex], and that the restriction of [itex]A[/itex] to [itex]V_2[/itex] is the identity [itex]1[/itex]?
[tex]
V_1+ V_2 = \mathbb{R}^N,\quad\quad \textrm{dim}(V_1)=2,\quad \textrm{dim}(V_2)=N-2
[/tex]
and such that the restriction of [itex]A[/itex] to [itex]V_1[/itex] belongs to [itex]\textrm{SO}(2)[/itex], and that the restriction of [itex]A[/itex] to [itex]V_2[/itex] is the identity [itex]1[/itex]?