Let W be a subspace of $\mathbb{R}^n$ with orthogonal basis $\left \{ \mathbf{w}_1 ,..., \mathbf{w}_p \right \}$ and let $\left \{ \mathbf{v}_1 ,..., \mathbf{v}_q \right \} $ be an orthogonal basis for $W^\perp$.
If I combine the two sets, that is, $\left \{ \mathbf{w}_1 ,..., \mathbf{w}_p ,\mathbf{v}_1 ,..., \mathbf{v}_q \right \} $ be an orthogonal set for $W^\perp$. Can I conclude that this set is an othogonal basis? Since they are all linearly independent.