Number of Rows (m) Enter the number of rows in your matrix (1 to 20)

Number of Columns (n) Enter the number of columns in your matrix (1 to 20)

Matrix A (Dimensionless) Enter your matrix values. Each row on a new line, values separated by spaces or commas. Supports decimals and negative numbers.

Quick Examples:

2x3 Matrix with Free Variables 3x3 Full Rank Matrix

Calculate Clear

This calculator is for informational purposes only. Verify results with appropriate professionals for important decisions. Numerical precision may vary for near-singular matrices.

What Is Null Space

The null space of a matrix is the set of all vectors that become zero when you multiply them by that matrix. Think of it as all the solutions to the equation Ax = 0, where A is your matrix and x is a vector. The null space tells you about the "hidden" directions where the matrix has no effect. It is also called the kernel of a linear transformation and plays a key role in understanding linear systems and their solutions.

How Null Space Is Calculated

Formula

The calculator uses Gaussian elimination to reduce your matrix to Reduced Row Echelon Form (RREF). It then identifies pivot columns and free variables. Each free variable gets assigned a parameter, and the calculator builds one basis vector for each free variable. The number of basis vectors equals the nullity, which is the number of columns minus the rank.

Why Null Space Matters

Understanding the null space helps you solve linear systems, check if a matrix is invertible, and analyze linear transformations. It connects directly to the fundamental theorem of linear algebra.

Why Null Space Is Important for Solving Linear Systems

When solving a system of linear equations, the null space shows you all the ways the solution can vary without changing the right-hand side. If the null space contains only the zero vector, your solution is unique. If it has many vectors, you have infinitely many solutions to choose from.

For Engineering Applications

Engineers use the null space to find equilibrium states in structures, analyze electrical circuits, and study mechanical systems. The null space reveals forces or movements that create no net effect on the system.

For Data Science and Machine Learning

In data science, the null space helps with dimensionality reduction and understanding feature relationships. Matrices with large null spaces have redundant information that can be compressed or removed without losing important patterns.

Example Calculation

Consider a student working with a 2x3 matrix from a homework problem. The matrix has values: first row is 1, 2, 3 and second row is 2, 4, 6. The student wants to find all vectors x such that Ax equals zero.

The calculator first converts this matrix to RREF form. The second row becomes all zeros because it is a multiple of the first row. This leaves one pivot column (the first column) and two free variables (second and third columns). Setting each free variable to 1 while the other is 0, the calculator constructs two basis vectors.

Rank = 1, Nullity = 2, Basis vectors: v1 = (-2, 1, 0) and v2 = (-3, 0, 1)

The result means any vector in the null space can be written as a combination of these two basis vectors. The student now knows that this matrix has a 2-dimensional null space, which explains why the homogeneous system has infinitely many solutions.

Frequently Asked Questions

Who is this Null Space Calculator for?

This calculator is designed for linear algebra students, engineering professionals, and researchers who need to analyze homogeneous linear systems. It works well for matrices up to 20 by 20 in size.

What does it mean when the null space is only the zero vector?

When the null space contains only the zero vector, the matrix has full column rank. This means the columns are linearly independent and the only solution to Ax = 0 is x = 0. The matrix is injective (one-to-one) as a linear transformation.

How is nullity related to the rank of a matrix?

The nullity plus the rank always equals the number of columns in your matrix. This is called the Rank-Nullity Theorem. If you know the rank, you can find the nullity by subtracting the rank from the number of columns.

Can I use this calculator for complex number matrices?

This calculator works with real number matrices only. For complex matrices, the null space computation follows similar steps but requires complex arithmetic. Consider using specialized software for complex matrix operations.

Why do I get different basis vectors than my textbook?

Basis vectors are not unique. Any set of linearly independent vectors that span the null space is valid. Different calculation methods or parameter choices can produce different but equivalent basis vectors. Both sets describe the same null space.