Orthogonal Subspaces I am reading orthogonality in subspaces and ran into confusion by reading this part:  
Suppose S is a six-dimensional subspace of nine-dimensional space $\mathbb R^9$.
a) What are the possible dimensions of subspace orthogonal to $S$? Answer: Sub spaces orthogonal to S can have dimensions $0,1,2,3.$
b) What are the possible dimensions of the orthogonal complement $S^{\perp}$ of $S$? Answer: Complement $S^{\perp}$ is the largest orthogonal subspace with dim $3$.
Where I am having trouble is understanding how the answers make sense to the question, or how the answers are pretty much the answers. In other words, for a, how is the dimensions $0,1,2,3$? But maybe I am not understanding the question. Any assistance with helping me understand the answer would be appreciated. 

Take all the vectors linearly independent vectors in $S$ and put them in a matrix (as rows). Since $S$ has dim=6, so there are 6 linearly independent vectors in S. Thus the matrix will have size 6x9. Now, rank of this matrix is $6$, and the orthogonal complement to $S$ is the rank of its null space. So, by rank nullity theorem, dimension of orthogonal complement is $3$. 
Now, subspaces orthogonal to $S$ consists of vectors that belong to the null space of the above matrix, and are subspaces of the null space of the matrix (any subspace will do). These subspaces can only have dimension $0,1,2$ or $3$. 
Hope this helps.
