Advanced Null Space Calculator

Enter Matrix Details

Choose the matrix size, adjust precision settings, then build or solve the matrix. The page uses a single-column layout with responsive input groups.

Rows

Columns

Display Precision

Pivot Tolerance

Matrix Entries

Example Data Table

This sample shows a 3 × 4 matrix with rank 2 and nullity 2.

Example	Matrix	Rank	Nullity	Sample Basis
Kernel Example	[1, 2, -1, 0] [2, 4, -2, 0] [0, 0, 1, -3]	2	2	[-2, 1, 0, 0], [3, 0, 3, 1]

Formula Used

The calculator solves the homogeneous system A·x = 0. It transforms the input matrix into reduced row echelon form, identifies pivot columns, and treats non-pivot columns as free variables.

Null Space(A) = { x : A·x = 0 } Nullity(A) = Number of Columns − Rank(A) Rank(A) + Nullity(A) = n

For each free variable, the calculator sets that variable to 1, all other free variables to 0, then solves the pivot variables from the reduced system. Those solution vectors form a basis for the null space.

How to Use This Calculator

Enter the number of rows and columns for your matrix.
Set display precision and the pivot tolerance if needed.
Click Build Matrix if you changed the size.
Fill every matrix entry with integers or decimals.
Press Calculate Null Space to see the result section above the form.
Review rank, nullity, pivot columns, the RREF matrix, row steps, and basis vectors.
Use the CSV and PDF buttons to export the completed analysis.

Frequently Asked Questions

1) What is the null space of a matrix?

The null space is the set of all vectors x that satisfy A·x = 0. It measures which inputs are sent to the zero vector by the matrix transformation.

2) What does nullity mean?

Nullity is the dimension of the null space. It tells you how many independent directions produce the zero output when multiplied by the matrix.

3) Why are pivot columns important?

Pivot columns identify leading variables in the reduced system. They determine the rank and show which columns are not free when solving A·x = 0.

4) What are free variables?

Free variables correspond to non-pivot columns. You can assign them parameters, then solve the pivot variables in terms of those parameters to describe the null space.

5) Can the null space contain only the zero vector?

Yes. That happens when the matrix has full column rank. In that case, the only solution to A·x = 0 is the trivial zero vector.

6) Does rank plus nullity always equal the number of columns?

Yes. This is the rank-nullity theorem. It is one of the main checks used when verifying a null-space calculation.

7) Can I use decimals instead of integers?

Yes. The calculator accepts decimal entries and uses a tolerance value to treat very small numbers as zero during row reduction.

8) Why export the results?

Exports help with homework records, teaching notes, engineering documentation, and study reviews. You can keep both numeric summaries and the derived basis vectors.