Null Space (Kernel) Calculator

About the Null Space (Kernel) Calculator

The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It is a subspace of Rⁿ and its dimension, called the nullity, measures the "degrees of freedom" in the homogeneous system. Together with the rank, the nullity satisfies the rank-nullity theorem: rank + nullity = n.

Finding the null space involves reducing the matrix to reduced row echelon form (RREF) and identifying the free variables — columns without pivots. Each free variable generates one basis vector for the null space. The basis vectors are constructed by setting each free variable to 1 (and others to 0) and solving for the pivot variables using back-substitution from the RREF.

The null space has profound implications. If it is trivial (only the zero vector), the system Ax = b has at most one solution for every b, and the columns of A are linearly independent. If the null space is non-trivial, there are infinitely many solutions to Ax = 0, and the general solution to Ax = b (when consistent) has the form x = x_particular + null(A).

This calculator computes the complete null space basis via RREF, verifies each basis vector by checking Ax = 0, and visualizes the results with component bars and column classification displays.

When This Page Helps

Finding the null space requires reducing the matrix to RREF, identifying free columns, and expressing each basis vector in terms of free parameters — a multi-step process where a single row-operation error yields the wrong dimension or wrong basis. This calculator handles matrices up to 4×4 with 7 presets covering singular, underdetermined, and rank-deficient cases. It shows every RREF step, highlights pivot vs. free columns, verifies each basis vector satisfies Ax = 0, and reports rank, nullity, and the full four fundamental subspaces. Essential for understanding solution sets.

How to Use the Inputs

1. Set the matrix dimensions and enter elements
2. Use preset buttons for common examples (singular, underdetermined, etc.)
3. View the nullity and basis vectors in the output
4. Check the column classification (pivot vs free) visualization
5. Verify that each basis vector satisfies Ax = 0
6. Toggle RREF steps to see the full reduction process

Example Calculation

Tips & Best Practices

• A square matrix has trivial null space if and only if it is invertible
• The null space dimension equals the number of free variables in RREF
• Every solution to Ax = b can be written as x = x_p + v where v is in the null space
• The null space of Aᵀ is called the left null space of A
• For wide matrices (more columns than rows), the null space is always non-trivial

How to Find the Null Space

The null space of A is the set of all solutions to Ax = 0. To find it, row-reduce A to RREF, then identify **pivot columns** (linearly independent) and **free columns** (dependent). For each free variable, set it to 1 and all other free variables to 0, then solve for the pivot variables by back-substitution. Each configuration yields one **basis vector** of the null space. The number of basis vectors equals the nullity = n − rank(A) for an m×n matrix.

The Rank-Nullity Theorem and the Four Fundamental Subspaces

The **rank-nullity theorem** states rank(A) + nullity(A) = n (number of columns). This partitions the domain ℝⁿ into the row space (dimension = rank) and the null space (dimension = nullity), which are orthogonal complements. The range (column space, dimension = rank) and the left null space (null(Aᵀ), dimension = m − rank) complete the four fundamental subspaces introduced by Gilbert Strang. Every vector in ℝⁿ splits uniquely into a row-space component and a null-space component.

Applications of the Null Space

In **linear systems**, the null space captures all the “freedom”: if x_p solves Ax = b, then every solution is x_p + v for v in null(A). In **control theory**, the null space of the observability matrix identifies unobservable states. In **computer graphics**, the null space of a homography matrix yields the epipole. In **machine learning**, null-space projection removes bias components or enforces constraints during training. A matrix with a trivial null space (nullity = 0) is injective — it maps distinct inputs to distinct outputs.

Frequently Asked Questions

• What is the null space of a matrix?

  The null space (kernel) is the set of all vectors x such that Ax = 0. It always includes the zero vector and may contain infinitely many vectors if the matrix is rank-deficient.

• How do I find the null space?

  Reduce the matrix to RREF, identify free (non-pivot) columns, then for each free variable set it to 1 and solve for the pivot variables. Each gives one basis vector.

• What does nullity tell us?

  Nullity is the dimension of the null space — the number of free parameters in the solution to Ax = 0. By the rank-nullity theorem, nullity = n − rank.

• When is the null space trivial?

  The null space is trivial (only contains the zero vector) when rank = n (number of columns), meaning all columns are linearly independent and every variable is a pivot variable.

• How does the null space relate to solutions of Ax = b?

  If x_p is any particular solution to Ax = b, then the general solution is x = x_p + null(A). The null space captures all the "freedom" in the solution set.

• What is the left null space?

  The left null space is the null space of Aᵀ — all vectors y such that Aᵀy = 0, or equivalently yᵀA = 0. It has dimension m − rank for an m×n matrix.