Null Space Calculator

Category: Linear Algebra

- April 16, 2025

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Calculate the null space (kernel) of a matrix. This calculator finds the basis vectors that span the null space and performs related calculations including rank, nullity, and row reduction.

Matrix Input

Number of Rows:

Number of Columns:

Enter Matrix Values:

Display Options

Decimal Places:

Number Format:

Show calculation steps

About Null Space

The null space (or kernel) of a matrix is a fundamental concept in linear algebra. It represents the set of all vectors that, when multiplied by the matrix, yield the zero vector. Understanding the null space helps solve homogeneous systems of linear equations and provides insights into the structure of linear transformations.

How to Find the Null Space

To find a basis for the null space of matrix A:

Convert the matrix to reduced row echelon form (RREF)
Identify free variables (columns without leading 1's)
Express the basic variables in terms of the free variables
Write the general solution as a linear combination of vectors
Extract the coefficient vectors to form a basis for the null space

Applications

The null space has important applications in:

Solving homogeneous systems of linear equations
Determining if a linear transformation is one-to-one (injective)
Finding the general solution to non-homogeneous systems
Analyzing differential equations and circuit networks
Computing eigenspaces in spectral theory
Image and signal processing (in finding essential information)

What Is the Null Space Calculator?

The Null Space Calculator is an interactive online tool that helps you find the null space of a matrix. The null space, also known as the kernel, is the set of all solutions \( \mathbf{x} \) to the equation \( A\mathbf{x} = \mathbf{0} \), where \( A \) is a matrix and \( \mathbf{0} \) is the zero vector.

Formula:

\[ \text{Ker}(A) = \{ \mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0} \} \]

What This Calculator Can Do

Finds a basis for the null space (kernel) of a matrix
Computes the matrix’s rank and nullity
Displays both Row Echelon Form (REF) and Reduced Row Echelon Form (RREF)
Shows step-by-step calculations (optional)
Accepts custom matrix inputs or lets you use an example matrix

How to Use the Null Space Calculator

Set the number of rows and columns of your matrix.
Enter the values into the matrix fields.
Optional: Use the example matrix for a quick demo.
Choose your display preferences: decimals or fractions, how many decimal places, and whether to show steps.
Click the Calculate button.
View your results, including the basis vectors, rank, nullity, REF, RREF, and more.

Why the Null Space Matters

Understanding the null space of a matrix is crucial for working with linear equations. It helps you:

Solve homogeneous systems of linear equations
Understand whether a matrix transformation is one-to-one
Find the general solution to matrix equations
Analyze linear independence and vector spaces

Applications in Linear Algebra

The null space is not just an academic concept — it's used in many advanced calculations and fields:

Linear systems: often used with the Gauss-Jordan Elimination Calculator
Transformations: related to tools like the Diagonalize Matrix Calculator and QR Factorization Calculator
Matrix operations: complements tools such as the Matrix Inverse Calculator, Matrix Multiplication Calculator, and Matrix Subtraction Calculator
Signal and image processing, control systems, data analysis

Key Concepts and Relationships

Rank: Number of pivot columns in the matrix (non-zero rows in RREF)
Nullity: Dimension of the null space (number of free variables)
Rank-Nullity Theorem:

\[ \text{Rank}(A) + \text{Nullity}(A) = \text{Number of columns in } A \]

Frequently Asked Questions (FAQ)

What is a basis for the null space?

A basis is a set of vectors that span the null space. Every solution to \( A\mathbf{x} = \mathbf{0} \) can be written as a linear combination of these vectors.

What if the null space is trivial?

If the only solution to \( A\mathbf{x} = \mathbf{0} \) is the zero vector, the null space is said to be trivial. This means the matrix is full rank.

Is this related to finding the inverse of a matrix?

Yes. If a square matrix has a trivial null space, it means it's invertible. You can use the Matrix Inverse Calculator to explore this further.

What are pivot and free variables?

Pivot variables correspond to columns with leading ones in RREF. Free variables are those that can take any value — they define the null space directions.

Can I use this for non-square matrices?

Absolutely. This tool works for any m × n matrix where m and n range from 1 to 10.

Explore Other Matrix Tools

This calculator is part of a broader set of linear algebra tools. Try these next:

LU Decomposition Calculator — Break down a matrix using LU matrix factorization
Matrix Exponential Calculator — Compute the exponential of a matrix
Vector Addition Calculator — Add vectors quickly and accurately
Matrix Transpose Calculator — Flip rows and columns of your matrix
Pseudoinverse Calculator — Find the Moore-Penrose pseudoinverse

Summary

The Null Space Calculator makes it easier to understand and explore fundamental properties of matrices. Whether you're studying for an exam or solving real problems in engineering or science, this tool helps break down the steps and visualize your results clearly. Try it alongside other matrix solvers like the Matrix Division Calculator or the QR Factorization Calculator to expand your toolkit.