The Gram-Schmidt process is an algorithm for orthogonalizing a set of vectors in an inner product space, often used to convert a basis into an orthonormal basis. It is fundamental in numerical linear algebra, facilitating processes like QR decomposition and improving the stability of computations involving vectors.

Gram-Schmidt Process

Gram-Schmidt process

algorithm for orthogonalizing

set of vectors

An inner product space is a vector space equipped with an additional structure called an inner product, which allows for the definition of geometric concepts such as angles and lengths. This structure enables the generalization of Euclidean geometry to more abstract vector spaces, providing a foundation for various applications in mathematics and physics.

inner product space

basis into an orthonormal basis

Numerical Linear Algebra focuses on the development and analysis of algorithms for performing linear algebra computations efficiently and accurately, which are fundamental to scientific computing and data analysis. It addresses challenges such as stability, accuracy, and computational cost in solving problems involving matrices and vectors.

numerical linear algebra

QR Decomposition is a matrix factorization technique that expresses a matrix as the product of an orthogonal matrix Q and an upper triangular matrix R. It is widely used in numerical linear algebra for solving linear systems, eigenvalue problems, and least squares fitting due to its numerical stability and efficiency.

QR decomposition

stability of computations

Vectors are mathematical entities that have both magnitude and direction, commonly used to represent physical quantities such as force and velocity. They are fundamental in fields like physics, engineering, and computer graphics, providing a way to describe spatial relationships and transformations in multi-dimensional spaces.

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