1. What is Null Space (Kernel)?

The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It represents the solutions to the homogeneous system of linear equations associated with the matrix.

2. How Does the Calculator Work?

The calculator performs Gaussian elimination to find the basis vectors of the null space:

Steps:

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

3. Importance of Null Space

Applications: The null space is fundamental in linear algebra, used in solving systems of equations, understanding linear transformations, and in many areas of engineering and physics.

4. Using the Calculator

Tips: Enter matrix rows separated by newlines, with elements separated by spaces or commas. The matrix can be any size (m×n).

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 mean?
A: If the null space contains only the zero vector, the matrix has full column rank.

Q3: How is null space related to linear independence?
A: The dimension of the null space equals the number of linearly dependent columns.

Q4: Can a matrix have an empty null space?
A: No, every matrix has at least the zero vector in its null space.

Q5: What's the practical use of null space?
A: Used in solving differential equations, control theory, and understanding network flows.