Gram-Schmidt process

Given a basis \mathbf v_1,\dots,\mathbf v_p of a subspace W, the Gram-Schmidt process builds \mathbf u_1=\mathbf v_1 and, for each k\ge 2, \mathbf u_k=\mathbf v_k-\sum_{j<k}\operatorname{proj}_{\mathbf u_j}(\mathbf v_k), producing an orthogonal basis \{\mathbf u_1,\dots,\mathbf u_p\} of the same W with the same nested spans at every step. It's the orthogonal decomposition theorem applied once per vector.