Gram-Schmidt Process

A rigorous construction of orthogonal and orthonormal bases, with dependence checks and numerical cautions.

Orthogonal projection separates a vector into fitted and perpendicular components. A vector is decomposed into a component along a subspace and a right-angled residual. Basis arrows show how coordinates reconstruct the projected result.

Precise definition

Given independent vectors $v_1,\ldots,v_k$ in a real inner-product space, Gram–Schmidt constructs orthogonal vectors $u_1=v_1$ and $u_j=v_j-\sum_{i<j}\operatorname{proj}_{u_i}v_j$, where $\operatorname{proj}_{u}v=\langle v,u\rangle u/\langle u,u\rangle$. Normalizing $q_i=u_i/\|u_i\|$ gives an orthonormal basis.

Notation and mathematical language

Each $u_j$ lies in $\operatorname{span}(v_1,\ldots,v_j)$, and the first $j$ new vectors span the same subspace as the first $j$ originals. The input order changes the resulting orthogonal basis but not the final spanned subspace.

Conceptual picture

At each stage, subtract every component in directions already built. The residual is perpendicular to all earlier $u_i$. Orthogonality makes coordinates and projections independent one-dimensional calculations.

Fully worked example

Interpretation and application

Gram–Schmidt produces QR factorizations, least-squares coordinates, signal bases, and stable geometric decompositions. Orthogonality depends on the selected inner product, which may incorporate weights or units.