GramSchmidtorthonormalisatie

Gram-Schmidt orthogonalization is a process in linear algebra that converts a finite set of linearly independent vectors into an orthogonal (or orthonormal) set spanning the same subspace. Named after Jørgen Gram and Erhard Schmidt, it is fundamental for constructing orthogonal bases and is a key step in QR factorization, projections, and least-squares calculations.

Given a sequence of vectors v1, v2, ..., vk in an inner product space, the classical Gram-Schmidt algorithm builds an orthogonal sequence u1, u2, ..., uk as follows: set u1 = v1. For i from 2 to k, form w_i = v_i minus its projections onto the previously computed vectors: w_i = v_i − sum_{j=1}^{i−1} proj_{u_j}(v_i), where proj_{u_j}(v_i) = (⟨v_i, u_j⟩ / ⟨u_j, u_j⟩) u_j. The orthogonal vectors are then u_i = w_i. If an orthonormal set is desired, normalize each vector: e_i = u_i / ||u_i||.

Modified Gram-Schmidt is a numerically more stable variant that orthogonalizes against previously computed vectors in a sequential, stepwise manner, reducing accumulation of round-off error. In practice, numerical linear algebra often uses this variant, or alternative methods such as Householder reflections or Givens rotations to form QR factorizations.

Output is an orthogonal or orthonormal set {u1, ..., uk} spanning the same subspace as {v1, ..., vk}, provided the original vectors are linearly independent. The method is widely used in solving linear systems, computing projections, and stabilizing numerical computations in matrix factorizations.