1. What is 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 finds a basis for the null space by:

Where:

• \( A \) — Input matrix
• \( x \) — Solution vector
• \( 0 \) — Zero vector

Explanation: The calculator performs row reduction to find the special solutions that form a basis for the null space.

3. Importance of Null Space

Details: The null space helps understand the solutions to linear systems, determines matrix rank, and is fundamental in many applications including differential equations and computer graphics.

4. Using the Calculator

Tips: Enter your matrix with rows separated by newlines and columns separated by spaces. For example:
1 2 3
4 5 6
7 8 9

5. Frequently Asked Questions (FAQ)

Q1: What does the dimension of the null space represent?
A: The dimension (nullity) equals the number of free variables in the system Ax=0, which is n - rank(A).

Q2: How is null space related to linear independence?
A: A matrix has a non-trivial null space if and only if its columns are linearly dependent.

Q3: 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 products Ax.

Q4: Can the null space be empty?
A: No, it always contains at least the zero vector. A non-empty null space means there are non-zero solutions.

Q5: How is null space used in applications?
A: It's used in solving differential equations, computer graphics (for transformations), and analyzing networks.