1. What is Matrix Nullity?

The nullity of a matrix is the dimension of its null space (kernel), which represents the number of linearly independent solutions to the homogeneous equation \( A\mathbf{x} = \mathbf{0} \). It's a fundamental concept in linear algebra that helps understand the solution space of linear systems.

2. How Does the Calculator Work?

The calculator uses the nullity formula:

Where:

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

Explanation: The formula comes from the Rank-Nullity Theorem, which relates the dimensions of the column space and null space of a matrix.

3. Importance of Nullity Calculation

Details: Knowing the nullity helps determine the number of free variables in a system of linear equations, understand the solution space structure, and analyze linear transformations.

4. Using the Calculator

Tips: Enter the matrix rank (must be ≤ number of columns) and the number of columns. Both must be positive integers with rank ≤ columns.

5. Frequently Asked Questions (FAQ)

Q1: What does nullity = 0 mean?
A: Nullity of 0 means the matrix has a trivial null space (only the zero vector), indicating the columns are linearly independent.

Q2: Can nullity be greater than the number of columns?
A: No, nullity is always ≤ number of columns, as rank is always ≥ 0.

Q3: How is nullity related to solutions of Ax=b?
A: For a consistent system, the general solution is particular solution + null space vectors. Nullity gives the number of free variables.

Q4: What's the difference between nullity and rank?
A: Rank is the dimension of the column space (image), while nullity is the dimension of the null space (kernel).

Q5: Does nullity depend on the field?
A: Yes, nullity can change if you consider the matrix over different fields (e.g., real vs. complex numbers).