What is the purpose of the Gram-Schmidt process?

The Gram-Schmidt process is a method used to take a finite, linearly independent set of vectors and turn it into an orthonormal set. This process is important in various applications, such as simplifying problems in numerical linear algebra and orthogonal projections in vector spaces.

In detail, the Gram-Schmidt process works by iteratively taking each vector from the original set and removing any components that are in the direction of the previously processed vectors to ensure orthogonality. After obtaining a set of orthogonal vectors, each vector is then normalized (scaled to have a length of 1) to ensure they are orthonormal. This orthonormality is particularly beneficial in computations involving projections and least squares, as it simplifies many calculations.

The other possibilities do not accurately capture the essence of the Gram-Schmidt process. Finding eigenvalues pertains to analyzing the properties of matrices rather than manipulating vectors directly. Maximizing lengths doesn't relate to the orthogonalization process, and determining if a matrix is symmetric involves checking the equality of elements rather than transforming vectors into a different form. Thus, the primary purpose of the Gram-Schmidt process is indeed to convert a set of vectors into an orthonormal set.