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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.

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It is not at all hasty. Presumably you already know the result that $p+q=n$. Since $W$ and $W^\perp$ have only the zero-vector in common, and since the $v_i$ are orthogonal to the $w_j$, you end up with an orthogonal basis for $\mathbb{R}^n$.

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