orthogonal subspaces in $\mathbb R^2$ Here is the question:
Let W be a subspace of $R^n$ with an orthogonal basis {$w_1$,...,$w_p$}, and let {$v_1$,...,$v_q$} be and orthogonal basis for W$\perp$. Assume that both p >= 1 and q>=1.
(1) For the case when n=2, describe geometrically W and W$\perp$. In particular, what are the possible dimensions of these two subspaces?
(2) For the case when n=3.....
My confusion is because I was under the impression that an orthogonal basis must consist of at least two vectors, because one vector cannot be orthogonal alone. But, if W is a two dimension subspace, then how can W$\perp$ be in $R^2$? If W is 2 dimensions, then the only orthogonal subspace would be a line in $R^3$. It seems to me that if both W and W$\perp$ are an orthogonal basis for a subspace, they must each span 2 vectors and together reside in a higher dimension than $R^2$. I also know that the union set of these two subspaces must span $R^2$ by definition, which tells me that there must be a total of two orthogonal vectors. Can someone give me a hint on what I'm missing here?
Thanks

If W is a space of two dimensions then it is not a subspace of R^2 it is the whole space.
The space perpendicular to W is null.
