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 (rank) and null space (nullity) of a matrix to the number of columns in the matrix.

2. How Does the Calculator Work?

The calculator uses the Rank-Nullity Theorem:

Where:

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

Explanation: The theorem states that for any matrix A, the sum of its rank and nullity equals the number of its columns.

3. Importance of Rank and Nullity

Details: Rank and nullity are fundamental concepts in linear algebra that help determine the properties of linear transformations, solve systems of linear equations, and understand matrix properties.

4. Using the Calculator

Tips: Enter the rank of the matrix (must be ≤ number of columns) and the number of columns in the matrix. Both values must be non-negative integers.

5. Frequently Asked Questions (FAQ)

Q1: What is the rank of a matrix?
A: The rank is the dimension of the vector space generated by its columns (column space) or rows (row space).

Q2: What is the nullity of a matrix?
A: The nullity is the dimension of the kernel (null space) of the matrix, which consists of all vectors that the matrix maps to the zero vector.

Q3: Can the rank be greater than the number of columns?
A: No, the rank cannot exceed the number of columns or the number of rows in a matrix.

Q4: What does nullity = 0 mean?
A: A nullity of zero means the matrix has a trivial null space (only the zero vector), indicating the matrix is injective (one-to-one).

Q5: How is this theorem useful in applications?
A: It's used in solving systems of linear equations, analyzing linear transformations, and understanding the structure of matrices in various applications.