Orthogonal Basis for Column Space Calculator

Gram-Schmidt Process:

\[ \begin{aligned} \mathbf{u}_1 &= \mathbf{v}_1 \\ \mathbf{u}_2 &= \mathbf{v}_2 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2) \\ \mathbf{u}_3 &= \mathbf{v}_3 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_3) - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3) \\ &\vdots \\ \mathbf{u}_k &= \mathbf{v}_k - \sum_{j=1}^{k-1} \text{proj}_{\mathbf{u}_j}(\mathbf{v}_k) \end{aligned} \]

Orthogonal Basis for Column Space:

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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 performs the Gram-Schmidt process on the column vectors of the input matrix:

Where:

\( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \) — Original column vectors
\( \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n \) — Orthogonal basis vectors
\( \text{proj}_{\mathbf{u}}(\mathbf{v}) \) — Projection of \( \mathbf{v} \) onto \( \mathbf{u} \)

Explanation: The process works by successively subtracting the projection of each new vector onto the subspace formed by the previous orthogonal vectors.

3. Importance of Orthogonal Basis

Details: An orthogonal basis simplifies many linear algebra computations, including least squares problems, QR decomposition, and makes numerical algorithms more stable.

4. Using the Calculator

Tips: Enter your matrix with each row on a new line and space-separated columns. For example:
1 2 3
4 5 6
7 8 9

5. Frequently Asked Questions (FAQ)

Q1: What if my matrix has linearly dependent columns?
A: The Gram-Schmidt process will produce fewer orthogonal vectors than the original number of columns, indicating the column space's true dimension.

Q2: Is the result orthonormal?
A: This calculator produces an orthogonal basis. To make it orthonormal, you would need to normalize each vector by dividing by its norm.

Q3: How does this relate to QR factorization?
A: The Gram-Schmidt process is one method to compute the Q matrix in QR decomposition, where Q is orthogonal and R is upper triangular.

Q4: What numerical issues might occur?
A: With nearly linearly dependent vectors, the classical Gram-Schmidt process can lose orthogonality due to rounding errors. Modified Gram-Schmidt is more stable.

Q5: Can I use this for function spaces?
A: The Gram-Schmidt process works in any inner product space, but this calculator is specifically for finite-dimensional vector spaces.