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 subspace in linear algebra that reveals the linear dependencies among the columns of A.

2. How to Calculate Null Space

The null space can be found by solving the homogeneous system Ax = 0 using Gaussian elimination:

Steps to compute:

1. Form the augmented matrix [A | 0]
2. Perform row reduction to reduced row echelon form (RREF)
3. Identify pivot and free variables
4. Express basic variables in terms of free variables
5. The free variables form the basis for the null space

3. Importance of Null Space

Applications: The null space is crucial for understanding solutions to linear systems, linear transformations, eigenvalues, and many areas of applied mathematics and engineering.

4. Using the Calculator

Instructions: Enter your matrix with rows separated by newlines and elements separated by spaces. The calculator will perform the computation and display a basis for the null space.

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 products Ax.

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

Q3: How is nullity related to rank?
A: The Rank-Nullity Theorem states: rank(A) + nullity(A) = number of columns in A.

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

Q5: How is null space used in applications?
A: It's used in solving differential equations, control theory, and understanding linear transformations.