1. What is Matrix Nullity?

Matrix nullity represents the dimension of the null space of a matrix, which is the number of linearly independent solutions to the homogeneous equation Ax = 0. It provides important information about the structure and properties of a linear transformation.

2. How Does the Calculator Work?

The calculator uses the fundamental nullity formula:

Where:

• \( n \) — Number of columns in the matrix
• \( \text{rank}(A) \) — Rank of the matrix (number of linearly independent rows/columns)

Explanation: This formula is derived from the Rank-Nullity Theorem, which states that for any matrix A, the sum of the rank and nullity equals the number of columns.

3. Importance of Nullity Calculation

Details: Calculating nullity is essential for understanding the solution space of linear systems, determining if a matrix is invertible, and analyzing the properties of linear transformations in various applications including engineering, physics, and computer science.

4. Using the Calculator

Tips: Enter the number of columns (n) as a positive integer and the matrix rank as a non-negative integer that does not exceed n. Both values must be valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What does nullity = 0 indicate?
A: Nullity = 0 indicates that the matrix has full column rank and the only solution to Ax = 0 is the trivial solution (x = 0).

Q2: Can nullity be negative?
A: No, nullity is always a non-negative integer since it represents the dimension of a vector space.

Q3: What is the relationship between nullity and invertibility?
A: A square matrix is invertible if and only if its nullity is 0 (and its rank equals its dimension).

Q4: How is nullity related to the solution of linear systems?
A: The nullity gives the number of free variables in the solution of the homogeneous system Ax = 0.

Q5: Can nullity exceed the number of columns?
A: No, nullity cannot exceed the number of columns n, since rank(A) ≥ 0 and nullity = n - rank(A).