GramSchmidtin

Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space. It is named after the mathematicians Erhard Schmidt and Joseph Louis Lagrange, who independently developed the process. The process takes a finite, linearly independent set of vectors and produces an orthogonal (or orthonormal) set of vectors that spans the same subspace.

The Gram-Schmidt process works by constructing each vector in the orthogonal set sequentially. For a given set of vectors {v1, v2, ..., vn}, the process generates an orthogonal set {u1, u2, ..., un} such that the span of {u1, u2, ..., uk} is the same as the span of {v1, v2, ..., vk} for each k. The first vector u1 is simply v1. For each subsequent vector uk, the process subtracts the projection of uk onto the subspace spanned by the previous vectors {u1, u2, ..., uk-1}.

The Gram-Schmidt process is widely used in various fields of mathematics and science, including linear algebra, numerical analysis, and machine learning. It is particularly useful in algorithms that require orthogonal or orthonormal bases, such as the QR decomposition and the singular value decomposition. The process is also used in the construction of orthogonal polynomials and in the solution of linear systems of equations.