1. What is the Gram-Schmidt Process?

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:

• \( u_i \) — Original vectors
• \( v_j \) — Orthogonal vectors
• \( \text{proj}_{v_j}(u_i) \) — Projection of \( u_i \) onto \( v_j \)

Explanation: Each new orthogonal vector is created by subtracting from the original vector its projections onto all previously created orthogonal vectors.

3. Importance of Orthogonalization

Details: Orthogonal vectors are fundamental in many areas of mathematics and engineering, including QR decomposition, signal processing, and solving systems of linear equations.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What happens if the input vectors are linearly dependent?
A: The process will produce a zero vector for any vector that is linearly dependent on the previous ones.

Q2: Are the resulting vectors orthonormal?
A: The results are orthogonal but not necessarily normalized. You can normalize them by dividing each by its magnitude.

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

Q4: Can I use this for complex vectors?
A: This calculator handles real vectors only. Complex vectors require conjugates in the inner product.

Q5: What applications use Gram-Schmidt?
A: Applications include least squares approximation, QR decomposition, and principal component analysis (PCA).