1. What is Image Basis?

The image (or range) of a matrix A is the set of all possible linear combinations of its column vectors. A basis for the image space consists of linearly independent vectors that span this space.

2. How Does the Calculator Work?

The calculator uses Singular Value Decomposition (SVD) to find the basis:

Where:

• \( A \) — Input matrix
• \( V \) — Right singular vectors from SVD of A
• \( r \) — Rank of the matrix (number of non-zero singular values)

Explanation: The right singular vectors corresponding to non-zero singular values form an orthonormal basis for the image space of A.

3. Importance of Image Basis

Details: Knowing the basis of the image space helps understand the output space of a linear transformation, solve linear systems, and perform dimensionality reduction.

4. Using the Calculator

Tips: Enter the matrix with each row on a new line, elements separated by spaces or commas. The rank parameter is optional - if not provided, it will be calculated automatically.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between image basis and null space?
A: The image is the output space of the transformation, while the null space contains inputs that map to zero.

Q2: Why use SVD for finding image basis?
A: SVD provides a numerically stable method that works for any matrix, including rank-deficient ones.

Q3: How does this relate to column space?
A: The image and column space are the same - the basis vectors are just orthonormalized.

Q4: Can I use this for non-square matrices?
A: Yes, the calculator works for any m×n matrix.

Q5: What if my matrix has complex numbers?
A: This calculator only handles real matrices. For complex matrices, the SVD approach still works but requires complex arithmetic.