1. What is Null Space?

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

2. How Does the Calculator Work?

The calculator finds the null space by:

Where:

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

Explanation: The calculator performs Gaussian elimination to find the basis vectors for the null space.

3. Importance of Null Space

Details: The null space is fundamental in linear algebra, helping understand linear transformations, solving systems of equations, and analyzing matrix properties like rank.

4. Using the Calculator

Tips: Enter your matrix with rows separated by newlines and elements separated by spaces. The matrix must be valid (rectangular with numeric values).

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: What does a zero null space indicate?
A: A zero null space (only the trivial solution x=0) means the columns of A are linearly independent.

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

Q5: How is null space used in applications?
A: Used in solving differential equations, computer graphics, data analysis, and more.