Gram-Schmidt Orthonormalization Calculator
This calculator helps you compute the orthonormal basis of a set of vectors using the Gram-Schmidt process. Ideal for students and professionals working with linear algebra.
Calculator
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Data Source and Methodology
All calculations are based on the Gram-Schmidt process as defined in standard linear algebra textbooks.
The Formula Explained
\( \text{u}_k = \text{v}_k - \sum_{j=1}^{k-1} \text{proj}_{\text{u}_j}(\text{v}_k) \)
Glossary of Variables
- vk: The original vector in the set.
- uk: The orthogonal vector in the basis.
- projuj(vk): The projection of vk onto uj.
How It Works: A Step-by-Step Example
Given vectors (1, 1, 1), (2, 1, 0), and (5, 1, 3), the calculator will output their orthonormal set.
Frequently Asked Questions (FAQ)
What is the Gram-Schmidt process?
It's a method for orthonormalizing a set of vectors in an inner product space.
Why use orthonormal vectors?
Orthonormal vectors simplify operations like projections and rotations in vector spaces.
Can I use this calculator for any vectors?
Yes, as long as the vectors are linearly independent.
What happens with dependent vectors?
The process will fail as it requires linear independence for orthonormalization.
Is the Gram-Schmidt process efficient?
It is efficient for small to medium-sized vector sets but may not be optimal for very large datasets.