1. What is Gram-Schmidt Orthogonalization?

The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space. It takes a finite, linearly independent set of vectors and generates an orthogonal set that spans the same subspace.

2. How Does the Calculator Work?

The calculator implements the Gram-Schmidt process:

Where:

• \( v_i \) — The i-th orthogonal vector
• \( u_i \) — The i-th input vector
• \( \text{proj}_{v_j} u_i \) — Projection of \( u_i \) onto \( v_j \)

Explanation: For each vector, subtract its projection onto all previously computed orthogonal vectors to make it orthogonal to them.

3. Importance of Orthogonalization

Details: Orthogonalization is fundamental in linear algebra, used in QR decomposition, least squares problems, and creating orthonormal bases for vector spaces.

4. Using the Calculator

Tips: Enter vectors separated by semicolons, with components separated by commas. All vectors must be of the same dimension.

5. Frequently Asked Questions (FAQ)

Q1: What if my vectors are linearly dependent?
A: The algorithm will produce a zero vector for any linearly dependent input vector.

Q2: Can I normalize the results?
A: Yes, you can normalize each orthogonal vector by dividing by its norm to get an orthonormal set.

Q3: What's the computational complexity?
A: For m vectors of dimension n, it's O(m²n) operations.

Q4: Are there numerical stability issues?
A: Yes, the classical Gram-Schmidt process can suffer from loss of orthogonality due to rounding errors.

Q5: What alternatives exist?
A: Modified Gram-Schmidt and Householder transformations are more numerically stable alternatives.