Enter Matrix Values
Choose matrix size, enter values, and solve Ax = 0. You may use decimals, scientific notation, or fractions like 3/4.
Example Data Table
| Example | Matrix | Rank | Nullity | Typical Basis Insight |
|---|---|---|---|---|
| Dependent columns | [[1, 2, 3], [2, 4, 6], [1, 1, 1]] | 2 | 1 | One free variable creates one basis vector. |
| Identity matrix | [[1, 0, 0], [0, 1, 0], [0, 0, 1]] | 3 | 0 | Only the zero vector solves Ax = 0. |
| Wide matrix | [[1, 0, 2, 3], [0, 1, -1, 4]] | 2 | 2 | Two parameters span a plane in solution space. |
Formula Used
The null space contains every vector x that satisfies the homogeneous system Ax = 0. Here, A is the input matrix and x is the unknown column vector.
First, reduce A to reduced row echelon form using elementary row operations. Pivot columns identify dependent variables, while nonpivot columns become free variables.
Nullity formula: nullity(A) = number of columns - rank(A).
For each free variable, set that variable to 1 and the remaining free variables to 0. Then solve the pivot variables from the reduced system to build one basis vector.
The full null space is the span of all basis vectors found this way.
How to Use This Calculator
- Pick the number of rows and columns for your matrix.
- Enter each matrix value. Fractions and decimals both work.
- Adjust tolerance if your matrix uses very small values.
- Set display decimals for cleaner formatted output.
- Click Calculate Null Space to generate the result.
- Review the RREF, pivot columns, free variables, and basis vectors.
- Use the CSV button for spreadsheet-ready results.
- Use the PDF button for a clean report export.
Frequently Asked Questions
1. What does the null space represent?
It is the set of all vectors that become zero after multiplication by the matrix. These vectors reveal hidden dependencies among the matrix columns.
2. Why are free variables important?
Free variables generate the null space. Each free variable usually contributes one independent basis vector, so they determine the solution space dimension.
3. What is the difference between rank and nullity?
Rank counts pivot columns. Nullity counts free-variable directions. Together they satisfy the rank-nullity relationship for the total number of columns.
4. Can this calculator handle rectangular matrices?
Yes. It works for tall, square, and wide matrices within the supported size range. Null spaces are often especially interesting for wide matrices.
5. Why does an identity matrix have a trivial null space?
Every column is a pivot column in the identity matrix. That leaves no free variables, so only the zero vector satisfies Ax = 0.
6. What tolerance should I use?
Use a small tolerance when near-zero rounding errors are expected. For clean integer matrices, the default tolerance is usually appropriate.
7. Can I enter fractions like 1/3?
Yes. The calculator accepts plain fractions, decimals, and scientific notation. Fractions are converted automatically before row reduction begins.
8. What if nullity equals zero?
Then the null space is trivial. The only solution to Ax = 0 is the zero vector, and no nonzero basis vectors exist.