1. What is Nullity?

The nullity of a matrix is the dimension of its null space (kernel). It represents the number of linearly independent solutions to the homogeneous equation \( A\mathbf{x} = \mathbf{0} \).

2. How Does the Calculator Work?

The calculator uses the fundamental relationship:

Where:

• \( A \) — Input matrix
• \( n \) — Number of columns in matrix
• \( \text{rank}(A) \) — Rank of matrix A

Explanation: The nullity is calculated by finding the rank of the matrix and subtracting it from the number of columns.

3. Importance of Nullity

Details: Nullity helps determine the number of free variables in a system of linear equations and is fundamental in linear algebra applications like solving differential equations and computer graphics.

4. Using the Calculator

Tips: Enter your matrix with each row on a new line, elements separated by spaces or commas. The calculator will compute the nullity and display basis vectors for the null space.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between nullity and rank?
A: Rank is the dimension of the column space (number of linearly independent columns), while nullity is the dimension of the null space.

Q2: What does nullity = 0 mean?
A: Nullity = 0 means the matrix has a trivial null space (only the zero vector), indicating the columns are linearly independent.

Q3: Can nullity be greater than the matrix size?
A: No, nullity ≤ number of columns. The rank-nullity theorem ensures nullity + rank = number of columns.

Q4: How is nullity used in applications?
A: It's used in solving systems of equations, analyzing linear transformations, and in fields like control theory and quantum mechanics.

Q5: What's the relationship between nullity and invertibility?
A: A square matrix is invertible if and only if its nullity is 0 (i.e., its null space contains only the zero vector).