1. What is Zero Space (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 solutions to the homogeneous system of linear equations.

2. How to Calculate Null Space

The null space can be found by:

Steps:

1. Form the augmented matrix [A|0]
2. Perform Gaussian elimination to reduced row echelon form
3. Identify free variables
4. Express basis vectors in terms of free variables

3. Importance of Null Space

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

4. Using the Calculator

Instructions: Enter your matrix using commas to separate elements within a row and semicolons to separate rows. For example, "1,2,3;4,5,6" represents a 2×3 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 for different x.

Q2: When does a matrix have a non-trivial null space?
A: When the matrix has linearly dependent columns (i.e., when rank is less than number of columns).

Q3: 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.

Q4: How is null space related to linear independence?
A: If the null space contains only the zero vector, the columns of A are linearly independent.

Q5: Can the null space be empty?
A: No, it always contains at least the zero vector.