If $V$ and $W$ are subspaces of the same dimension such that $V$ meets $W^\perp$, then $W$ meets $V^\perp$ I'm having a little difficulty understanding the proof for orthogonal complements.  I kind of understand orthogonal complements, but I cannot seem to find a logic to this.  I'm trying to follow along with theorems and definitions, but I'm not far.  Any advice on how to go about his proof problem?

Let V and W be two subspace of $\mathbb{R}^n$ of the same dimension, $m$.  Assume $\exists \mathbf{v} \in V \setminus \left \{ 0 \right \}$ such that $\mathbf{v} \in W^{\bot}$.
(a) Prove that $\exists \mathbf{w} \in W \setminus \left \{ 0 \right \}$ such that $\mathbf{w} \in V^{\bot}$.
(b) Show by an example that the conclusion of part (a) may not hold if V and W do not have the same dimension.


Think about projection onto $V$. This defines a linear transformation $\mathbb{R}^n \rightarrow V$ whose kernel is precisely $V^\perp$. What are the dimensions of this projection's matrix, restricted to $W$? What does the given condition say about this matrix, if an appropriate basis for $V$ is chosen?
