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 its number of columns.

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
• \( \text{Number of Columns}(A) \) — Total number of columns in matrix A

Explanation: The theorem states that for any matrix A, the sum of the dimensions of its column space and null space equals the number of its columns.

3. Importance of the Theorem

Details: The Rank-Nullity Theorem is crucial for understanding linear transformations, solving systems of linear equations, and analyzing matrix properties. It helps determine if a system has solutions and how many solutions exist.

4. Using the Calculator

Tips: Enter any two known values to calculate the third. All values must be non-negative integers. The calculator will verify if the theorem holds when all three values are provided.

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: Does this theorem apply to non-square matrices?
A: Yes, the theorem applies to any m×n matrix, whether square or rectangular.

Q4: How is this related to solving linear systems?
A: For a system Ax=b, the number of free variables equals the nullity, while the number of pivot variables equals the rank.

Q5: What's the relationship with the dimension theorem?
A: The Rank-Nullity Theorem is a special case of the dimension theorem for linear transformations between vector spaces.