1. What is the Null Space?

The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It's a fundamental concept in linear algebra that reveals the solutions to homogeneous systems of linear equations.

2. How Does the Calculator Work?

The calculator performs Gaussian elimination to find the null space:

Steps:

1. Convert matrix to row echelon form
2. Identify pivot and free variables
3. Solve for basis vectors of the null space

3. Importance of Null Space Calculation

Applications: The null space is essential for understanding linear transformations, solving differential equations, and analyzing systems of linear equations.

4. Using the Calculator

Tips: Enter matrix values separated by commas for rows and semicolons for columns. For example: "1,2,3;4,5,6;7,8,9" for a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between null space and column space?
A: The null space contains solutions to Ax=0, while the column space contains all possible outputs Ax.

Q2: What does a zero null space indicate?
A: A zero null space (only the trivial solution) means the matrix has full column rank.

Q3: How is null space related to linear independence?
A: The dimension of the null space (nullity) indicates how many linearly dependent columns exist.

Q4: Can the null space be empty?
A: No, it always contains at least the zero vector.

Q5: What's the relationship between null space and eigenvalues?
A: The null space of (A - λI) gives eigenvectors for eigenvalue λ.