1. What is the Gram-Schmidt Process?

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

2. How Does the Calculator Work?

The calculator implements the Gram-Schmidt process:

Where:

• \( u_i \) — Original input vectors
• \( v_j \) — Orthonormal vectors already computed
• \( \text{proj}_{v_j} u_i \) — Projection of \( u_i \) onto \( v_j \)

Explanation: The process subtracts from each new vector its projections onto all previously computed orthonormal vectors, then normalizes the result.

3. Importance of Orthonormalization

Details: Orthonormal bases are fundamental in linear algebra, signal processing, quantum mechanics, and computer graphics. They simplify calculations and provide numerical stability.

4. Using the Calculator

Tips: Enter vectors as comma-separated values, one vector per line. All vectors must have the same dimension and be linearly independent for proper results.

5. Frequently Asked Questions (FAQ)

Q1: What happens if vectors are linearly dependent?
A: The process will produce a zero vector when it encounters linear dependence, which cannot be normalized.

Q2: Can I use this for complex vectors?
A: This calculator handles real vectors only. For complex vectors, the inner product definition changes.

Q3: What's the difference between orthogonal and orthonormal?
A: Orthogonal vectors are perpendicular, orthonormal vectors are both perpendicular and unit length.

Q4: How accurate are the results?
A: Results are accurate to 4 decimal places, but numerical precision limitations may affect very small values.

Q5: What applications use Gram-Schmidt orthonormalization?
A: QR decomposition, solving linear systems, principal component analysis, and many numerical algorithms.