Intersection of Subspaces with $O(3)$ Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below.
For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three matrices, consider its intersection with $O(3)$.


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*Is it generically true that $V \cap O(3)$ is non-empty (i.e. the set of such $V$ is an open subset of $\mathrm{Gr}(3, 6)$)? Can one say how many points the intersection generically includes?

*Is it possible to characterize the $6$-dimensional subspaces $V$ such that $V \cap O(3) = \emptyset$? (For example, if $V$ consists only of singular matrices, then $V \cap O(3) = \emptyset$; conversely, does every $V$ with $V \cap O(3) = \emptyset$ consist only of singular matrices?)



Old thread Let $V$ be a $6$-dimensional subspace of the space $\mathbb{R}^{3\times 3}$ of real three-times-three matrices. Is the intersection of $V$ with $O(3) \subset \mathbb{R}^{3 \times 3}$ always non-empty?
If not, how can we characterize those subspaces $V$ such that $V \cap O(3) = \emptyset$?
\Edit: Anton made the trivial comment below that one can take matrices with first (or some other) column equal to zero. However, how can we characterize such subspaces?
 A: In general the answer is no. For example, the matrices whose first column is a column of zeroes form a $6-$dimensional subspace not containing any ortogonal matrix, because it does not contain any invertible matrix.
A: The answer to your first question is 'yes, generically, the intersection is finite and transverse'. More precisely, the set $U$ of 6-dimensional subspaces that intersect $\mathrm{SO}(3)$ transversely (and hence in a finite number of points), is a dense open set in $\mathrm{Gr}(6,9)$. (However, $U$ is not connected; the number of intersection points can be (and are) different for different components of $U$.)  For example, the $6$-dimensional subspace $V_+$ consisting of upper triangular matrices intersects $\mathrm{SO}(3)$ transversely in the $4$ points that consist of the diagonal elements of $\mathrm{SO}(3)$.  Thus every $6$-dimensional subspace sufficiently near $V_+$ also intersects $\mathrm{SO}(3)$ in $4$ points and transversely.
As for your parenthetical question, the answer to that is 'no, there exist $6$-dimensional subspaces $W$ that do not meet $\mathrm{SO}(3)$ and for which the generic element in $W$ is invertible'.  For example, the generic $W$ that is sufficiently near the $W_0$ consisting of the matrices with zero first column will have its generic element be invertible.
