1. What Are Matrix Rank and Nullity?

The rank of a matrix is the dimension of the vector space generated by its columns (column rank) or rows (row rank). The nullity is the dimension of the kernel (null space) of the matrix.

2. How Does the Calculator Work?

The calculator uses Python's numpy library to compute:

Where:

• \( A \) — Input matrix
• \( \text{np.linalg.matrix_rank} \) — Numpy function to compute matrix rank
• \( A.\text{shape}[1] \) — Number of columns in the matrix

3. Importance of Rank and Nullity

Details: The rank-nullity theorem is fundamental in linear algebra. Rank determines the number of linearly independent solutions, while nullity measures the dimension of the solution space for Ax=0.

4. Using the Calculator

Tips: Enter your matrix in Python numpy array format (e.g., [[1,2],[3,4]]). The calculator will execute the Python code to compute rank and nullity.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between rank and nullity?
A: According to the rank-nullity theorem: rank + nullity = number of columns.

Q2: What does rank tell us about a matrix?
A: Rank indicates the number of linearly independent rows or columns, and thus the dimension of the image of the linear transformation.

Q3: When is nullity zero?
A: Nullity is zero when the matrix has full column rank (its columns are linearly independent).

Q4: How does numpy compute matrix rank?
A: Numpy uses SVD (Singular Value Decomposition) and counts the number of singular values above a certain threshold.

Q5: Can I use this for non-square matrices?
A: Yes, the rank-nullity theorem applies to any m×n matrix.