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 represents all possible solutions to the homogeneous equation Ax = 0.

2. How Null Space is Calculated

The null space is found by:

Calculation steps:

• Perform Gaussian elimination to reduce matrix to row echelon form
• Identify pivot and free variables
• Express basis vectors in terms of free variables

3. Importance of Null Space

Applications: Null space is fundamental in linear algebra for understanding linear transformations, solving systems of equations, and analyzing matrix properties.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

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

Q2: How is null space related to rank?
A: Rank-Nullity Theorem: rank(A) + nullity(A) = number of columns in A.

Q3: Can the null space be empty?
A: No, it always contains at least the zero vector.

Q4: What does a trivial null space mean?
A: A null space containing only the zero vector indicates the columns of A are linearly independent.

Q5: How is null space used in applications?
A: Used in solving differential equations, computer graphics, machine learning, and analyzing network flows.