orthogonal decomposition theorem

For a subspace W of \mathbb R^n and any \mathbf y\in\mathbb R^n, the orthogonal decomposition theorem guarantees \mathbf y=\hat{\mathbf y}+\mathbf z in exactly one way, with \hat{\mathbf y}\in W (the orthogonal projection of \mathbf y onto W) and \mathbf z\in W^\perp. Both existence and uniqueness come directly from building \hat{\mathbf y} out of an orthogonal basis of W.