1. What is the Rank-Nullity Theorem?

The Rank-Nullity Theorem is a fundamental theorem in linear algebra that relates the dimensions of the column space and null space of a matrix. It states that for any matrix A, the sum of the rank (dimension of column space) and nullity (dimension of null space) equals the number of columns of A.

2. How Does the Calculator Work?

The calculator uses the Rank-Nullity Theorem:

Where:

• \( n \) — Number of columns in the matrix
• \( \text{rank} \) — Rank of the matrix (dimension of column space)
• \( \text{Nullity} \) — Dimension of the null space

Explanation: The theorem provides a direct relationship between the number of linearly independent columns and the dimension of the solution space for the homogeneous system Ax = 0.

3. Importance of Nullity Calculation

Details: Calculating nullity is essential for understanding the solution space of linear systems, determining the number of free variables, and analyzing the properties of linear transformations represented by matrices.

4. Using the Calculator

Tips: Enter the number of columns (n) and the rank of the matrix. The rank must be a non-negative integer and cannot exceed the number of columns.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between rank and nullity?
A: Rank and nullity are complementary - their sum always equals the number of columns in the matrix.

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

Q3: What does a nullity of zero indicate?
A: A nullity of zero means the matrix has full column rank and the homogeneous system Ax = 0 has only the trivial solution.

Q4: How is nullity related to linear independence?
A: The nullity represents the number of linearly dependent columns or the number of free variables in the system.

Q5: Does this theorem apply to non-square matrices?
A: Yes, the Rank-Nullity Theorem applies to any m × n matrix, where n is the number of columns.