nonorthonormal

Nonorthonormal refers to a set of vectors that is not orthonormal. In linear algebra, an orthonormal set consists of vectors that are mutually orthogonal and have unit length. A nonorthonormal set may fail one or both of these conditions: vectors may not be orthogonal, or may not have unit length (or both). The term is often used when discussing bases, frames, or collections of vectors that are used to represent other vectors but do not satisfy the orthonormal property.

If a nonorthonormal set {v1, ..., vk} forms a basis for a vector space, any vector x in that space can be written uniquely as x = sum_i c_i v_i. However, unlike an orthonormal basis, the coefficients c_i are not simply inner products ⟨x, v_i⟩. Instead, the Gram matrix G with entries g_{ij} = ⟨v_i, v_j⟩ is used: G c = b, where b_i = ⟨x, v_i⟩. If the vectors are linearly independent, G is invertible and c = G^{-1} b. The Gram matrix equals the identity only for an orthonormal basis.

Nonorthonormal sets influence projections and expansions: projecting x onto the span of the set requires solving a linear system rather than straightforward inner-product coefficients. Gram-Schmidt orthogonalization can convert a nonorthonormal basis into an orthonormal one, facilitating simple coordinates and projections, though this process changes the basis without altering the span.

Examples illustrate differences: in R^2, vectors (1,0) and (1,1) are nonorthonormal; their Gram matrix is [[1,1],[1,2]]. They span the plane, but coordinate computation and projections are more involved than with an orthonormal basis.