Gram-Schmidt orthogonalization is a mathematical procedure used to transform a set of linearly independent vectors into an orthogonal orthonormal set. This process is commonly employed in linear algebra and functional analysis.
In simpler terms, given a set of vectors, the Gram-Schmidt process constructs a new set of orthogonal vectors that span the same subspace. The orthogonalization involves successively subtracting the components of the preceding vectors from the current vector, ensuring orthogonality.
This technique is valuable in various applications, particularly in solving linear systems, approximating solutions, and constructing orthogonal bases. The resulting orthonormal set simplifies mathematical computations and aids in the analysis of vector spaces. Gram-Schmidt orthogonalization is a fundamental tool in linear algebra for creating orthonormal bases from arbitrary sets of vectors.
