Gram-Schmidt Orthonormalization Calculator
The Gram-Schmidt Orthonormalization Calculator is designed for linear algebra students and professionals to convert a set of vectors into an orthonormal set using the Gram-Schmidt process.
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Data Source and Methodology
Calculations are based on the standard mathematical approach to the Gram-Schmidt process. For further details, refer to [this source](https://www.example.com).
The Formula Explained
The Gram-Schmidt process takes a finite, linearly independent set S = {v1, ..., vk} for k ≤ n and generates an orthogonal set T = {u1, ..., uk} that spans the same k-dimensional subspace of R^n as S. The orthonormal set is obtained by normalizing T.
Glossary of Terms
- Vector: A quantity having direction as well as magnitude.
- Orthogonal: Vectors that are perpendicular to each other.
- Orthonormal: Orthogonal vectors of unit length.
How It Works: A Step-by-Step Example
For vectors (1, 1, 1), (2, 1, 0), and (5, 1, 3), the Gram-Schmidt process generates orthonormal vectors through iterative orthogonalization and normalization.
Frequently Asked Questions (FAQ)
What is the Gram-Schmidt process?
The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, typically the Euclidean space.
How does this calculator work?
Enter vectors, and the calculator applies the Gram-Schmidt process to produce orthonormal vectors.