Result
Enter a matrix and press Solve.
Null Space of a Matrix Calculator
Enter a numeric matrix, reduce it to RREF, and compute a basis for Null(A).
Matrix
InputFormula
Null(A) = { x : Ax = 0 }
nullity(A) = n - rank(A)
Steps
- Read A as an m x n matrix.
- Row-reduce A to RREF.
- Identify pivot and free columns.
- Solve Ax = 0.
- Build the null space basis.
RREF
RREF will appear here after solving.
Matrix Analysis
What the Null Space of a Matrix Calculator Helps You Find
A null space calculator turns row reduction into a clear result, showing the vectors that satisfy Ax = 0 and helping you understand the solution space behind a matrix.
Clear RREF Workflow
See how the matrix is reduced into row echelon form so the pivot columns, free variables, and dependent relationships are easier to follow.
Basis Vector Results
Get the null space expressed as a basis, which is the most useful format for homework, engineering work, and linear algebra review.
Homogeneous System Support
The calculator focuses on solutions to Ax = 0, making it ideal when you need to solve a homogeneous system without manually tracking every row operation.
Rank and Free Variable Insight
Understand how the rank of a matrix affects the number of free variables and why that determines the dimension of the null space.
Helpful Learning Companion
Use the result to check your own row-reduction work, compare basis vectors, and build confidence before exams or assignments.
Fast Error Checking
Quickly confirm whether a nonzero null space exists, whether the matrix has only the trivial solution, or whether a calculation needs another look.
Simple Process
How to Use the Null Space Calculator
Enter your matrix, review the reduction, and read the null space result in a form that is ready to use in your solution or notes.
01
Enter the Matrix Values
Start with the coefficients from your matrix. Keep each entry in the correct row and column so the row operations reflect the original system accurately.
02
Review the Row Reduction
The calculator identifies pivots and free variables by reducing the matrix. This step explains why the final null space has its particular dimension.
03
Use the Basis for Ax = 0
Read the final vectors as a basis for the null space. Any linear combination of those basis vectors is a solution to the homogeneous system.
Practical Uses
Where Null Space Results Are Useful
The null space is more than a classroom topic. It appears anywhere linear relationships, constraints, transformations, and systems of equations need to be understood.
Linear Algebra Homework
Check the basis of a null space, confirm your free variables, and make your written solution easier to explain step by step.
Homogeneous Equations
Find every solution to Ax = 0 by expressing the answer as a span of vectors instead of a long list of separate equations.
Vector Space Interpretation
Understand whether the null space forms a line, plane, higher-dimensional subspace, or only the zero vector.
Engineering Constraints
Analyze systems where certain outputs must equal zero, such as balanced forces, circuit relationships, or constrained design equations.
Dependence Detection
Use null space behavior to spot dependent columns and understand when a matrix transformation loses information.
Exam Preparation
Practice interpreting pivots, rank, nullity, and basis vectors so the underlying concepts feel less mechanical and more connected.
Reliable Support
A Cleaner Way to Check Matrix Work
Use the calculator as a fast, focused companion when you need accurate null space results without unnecessary friction.
Free and Accessible
Work through null space problems whenever you need to verify a result, review a concept, or prepare a solution without creating an account.
Built for Quick Review
The layout keeps the important details easy to scan, from the row-reduced matrix to the final basis vectors and nullity.
Works on Any Screen
Whether you are studying from a laptop, tablet, or phone, the content stays readable and the matrix explanation remains easy to follow.
