Gram-Schmidt
Orthonormalize two vectors in R^2 with the Gram-Schmidt process calculator. Computes the orthonormal basis from your input vectors with step-by-step results.
About This Calculator
The Gram-Schmidt process is a fundamental algorithm in linear algebra that converts a set of linearly independent vectors into an orthonormal basis -- a collection of vectors that are mutually orthogonal (perpendicular, with dot product equal to zero) and each of unit length (norm equal to one). This Gram-Schmidt calculator applies the orthogonalization process to two vectors in two-dimensional space R^2, producing an orthonormal basis that spans the same vector subspace as the original vectors.
Methodology
The algorithm proceeds in three steps: (1) Normalize v1 to obtain the first orthonormal basis vector u1 = v1 / ||v1||. (2) Orthogonalize v2 by subtracting its projection onto u1: w2 = v2 - (v2 . u1) u1, yielding a vector w2 orthogonal to u1. (3) Normalize w2 to obtain the second orthonormal basis vector u2 = w2 / ||w2||. The resulting pair {u1, u2} forms an orthonormal basis for the span of {v1, v2}.
Applications
Orthonormal bases are essential throughout applied mathematics and engineering. In computer graphics, they define camera coordinate systems and rotation matrices. In signal processing, the Gram-Schmidt idea underlies Fourier series and wavelet decomposition. In data science and machine learning, QR decomposition (built on Gram-Schmidt) powers least-squares regression and numerical linear algebra solvers. In quantum mechanics, orthonormal bases represent the possible states of a quantum system.
Regional Notes
The Gram-Schmidt process is a universal mathematical algorithm taught in linear algebra courses worldwide (IN, US, UK). It is not region-specific -- the underlying math is identical across all countries. Students and professionals everywhere use it for QR decomposition, eigenvalue computation, and solving least-squares problems.
Frequently Asked Questions
What is the Gram-Schmidt process?
The Gram-Schmidt process is an algorithm that converts a set of linearly independent vectors into an orthonormal basis -- vectors that are mutually orthogonal (perpendicular) and each of unit length (normalized). It is fundamental to linear algebra and underpins QR decomposition.
How does Gram-Schmidt orthogonalization work?
The algorithm takes each input vector, subtracts its projection onto all previously computed orthonormal vectors, and normalizes the result. For two 2D vectors: normalize the first vector to get unit vector 1, project v2 onto unit vector 1 and subtract to get w2 orthogonal to unit vector 1, then normalize w2 to get unit vector 2.
What does orthonormal mean?
Two vectors are orthonormal if they are orthogonal (their dot product equals zero, meaning they are perpendicular) and each has unit length (norm equals 1). An orthonormal basis is a set of such vectors that spans the space.
Can Gram-Schmidt handle linearly dependent vectors?
Yes, but when vectors are linearly dependent, one of the resulting orthogonal vectors becomes the zero vector and is discarded, reducing the dimension of the orthonormal basis. The calculator will warn you when linear dependence is detected.
Why is the Gram-Schmidt process important?
It produces an orthonormal basis which simplifies many linear algebra computations including QR decomposition, least-squares approximation in data science, eigenvalue algorithms, signal processing (Fourier series and wavelets), and quantum mechanics.
Is this Gram-Schmidt calculator free to use?
Yes, all calculators on Calculy are completely free to use with no registration or limits.
What is QR decomposition and how is it related?
QR decomposition factors a matrix A into Q (an orthogonal matrix whose columns are produced by the Gram-Schmidt process) and R (an upper triangular matrix). It is widely used to solve linear systems and least-squares problems in numerical linear algebra.
Can I use this calculator for 3D vectors?
This calculator currently supports 2D vectors in R^2. The Gram-Schmidt process extends naturally to any number of dimensions -- for each additional coordinate, the same orthogonalization and normalization steps apply component-wise.