Nullspace Calculator

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Nullspace Calculator – Find the Basis for the Nullspace


Nullspace Calculator

Easily find the nullspace, nullity, and rank of a 3×3 matrix with our Nullspace Calculator.

Matrix Elements (3×3)

Enter the elements of your 3×3 matrix A:



Chart showing components of nullspace basis vectors (if 1D, 2D, or 3D).

What is a Nullspace Calculator?

A Nullspace Calculator is a tool used in linear algebra to find the nullspace (also known as the kernel) of a given matrix. The nullspace of a matrix A consists of all vectors x such that when A is multiplied by x, the result is the zero vector (Ax = 0). This calculator determines the basis vectors that span the nullspace, as well as the nullity (dimension of the nullspace) and the rank of the matrix.

Anyone studying or working with linear algebra, such as students, engineers, scientists, and mathematicians, can use a Nullspace Calculator. It helps in understanding the properties of linear transformations and solving systems of linear homogeneous equations.

A common misconception is that the nullspace is always just the zero vector. While the zero vector is always part of the nullspace, it is only the *entire* nullspace if the matrix columns are linearly independent (and the matrix is square or tall, resulting in full column rank).

Nullspace Formula and Mathematical Explanation

To find the nullspace of a matrix A, we solve the homogeneous system of linear equations Ax = 0. This is typically done by performing Gaussian elimination (row reduction) on the matrix A to transform it into its Reduced Row Echelon Form (RREF).

The steps are:

  1. Write down the matrix A.
  2. Apply elementary row operations to transform A into its RREF.
  3. Identify the pivot columns (columns with leading 1s in RREF) and free columns (columns without leading 1s).
  4. The variables corresponding to pivot columns are pivot variables, and those corresponding to free columns are free variables.
  5. Express the pivot variables in terms of the free variables using the equations derived from the RREF.
  6. For each free variable, set it to 1 and all other free variables to 0, and solve for the pivot variables. This gives a basis vector for the nullspace. Repeat for each free variable.
  7. The collection of these vectors forms a basis for the nullspace of A. The number of vectors in the basis is the nullity of A.

The Rank-Nullity Theorem states: Rank(A) + Nullity(A) = number of columns in A. Our Nullspace Calculator uses this principle.

Variable Meaning Unit Typical Range
A The input matrix Matrix elements Real numbers
x Vector in the domain Vector components Real numbers
0 The zero vector Vector components 0
RREF(A) Reduced Row Echelon Form of A Matrix elements 0 or 1 for pivots
Rank(A) Number of pivot columns in RREF(A) Integer 0 to min(rows, cols)
Nullity(A) Number of free columns in RREF(A) Integer 0 to cols

Practical Examples (Real-World Use Cases)

Example 1: A 2×3 Matrix

Consider the matrix A = [[1, 2, 3], [2, 4, 6]]. Let’s find its nullspace using the principles of our Nullspace Calculator.

The RREF of A is [[1, 2, 3], [0, 0, 0]].
Pivot column: 1 (x1)
Free columns: 2 (x2), 3 (x3)
Equation from RREF: x1 + 2×2 + 3×3 = 0 => x1 = -2×2 – 3×3

If x2=1, x3=0 => x1=-2. Vector: [-2, 1, 0]T
If x2=0, x3=1 => x1=-3. Vector: [-3, 0, 1]T

The basis for the nullspace is {[-2, 1, 0]T, [-3, 0, 1]T}. Nullity = 2, Rank = 1.

Example 2: A 3×3 Matrix with Nullity 1

Let A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. Using the Nullspace Calculator‘s method, the RREF is [[1, 0, -1], [0, 1, 2], [0, 0, 0]].
Pivots: x1, x2; Free: x3.
x1 – x3 = 0 => x1 = x3
x2 + 2×3 = 0 => x2 = -2×3

Set x3=1 => x1=1, x2=-2. Basis vector: [1, -2, 1]T. Nullity = 1, Rank = 2.

How to Use This Nullspace Calculator

  1. Enter Matrix Elements: Input the values for each element of the 3×3 matrix A into the corresponding fields (A(1,1) to A(3,3)).
  2. Calculate: The calculator automatically updates as you type, or you can click “Calculate Nullspace”.
  3. View Results: The calculator displays the nullity, the basis vectors for the nullspace, the rank, and the RREF of the matrix.
  4. Interpret Basis: The basis vectors are a set of linearly independent vectors that span the nullspace. Any vector in the nullspace can be written as a linear combination of these basis vectors.
  5. Check RREF: The RREF table shows the matrix after row reduction, which is used to find the nullspace.
  6. Use Chart: The chart visualizes the components of the basis vectors if they are in 1D, 2D, or 3D.
  7. Reset: Click “Reset” to clear the inputs to default values.
  8. Copy: Click “Copy Results” to copy the findings to your clipboard.

Understanding the nullspace helps determine if a system Ax=b has unique solutions and the structure of the solution set for Ax=0.

Key Factors That Affect Nullspace Results

  • Matrix Elements: The specific values within the matrix directly determine its RREF and thus its nullspace. Small changes can alter the rank and nullity.
  • Number of Rows and Columns: The dimensions of the matrix constrain the maximum possible rank and influence the nullity (via Rank-Nullity Theorem). Our current Nullspace Calculator is for 3×3.
  • Linear Independence of Columns: If the columns are linearly independent, the nullspace contains only the zero vector (nullity=0 for square/tall matrices with full column rank).
  • Linear Independence of Rows: This determines the rank of the matrix, which in turn affects the nullity.
  • Rank of the Matrix: The rank is the number of linearly independent rows or columns (or pivot positions in RREF). Nullity = number of columns – rank.
  • Presence of Free Variables: The number of free variables after row reduction equals the nullity and determines the number of basis vectors for the nullspace. Each free variable corresponds to a basis vector.

Frequently Asked Questions (FAQ)

What is the nullspace of a matrix?
The nullspace (or kernel) of a matrix A is the set of all vectors x for which Ax = 0. It’s a subspace of the domain.
What is nullity?
Nullity is the dimension of the nullspace, which is the number of vectors in any basis for the nullspace. It equals the number of free variables in the system Ax=0.
How does the Nullspace Calculator find the basis?
It row-reduces the matrix to RREF, identifies pivot and free variables, and then expresses pivot variables in terms of free variables to construct basis vectors.
What if the nullity is 0?
If the nullity is 0, the nullspace contains only the zero vector, and the columns of the matrix are linearly independent (if it’s square or tall).
Can I use this Nullspace Calculator for non-square matrices?
This specific calculator is designed for 3×3 matrices. The concept of nullspace applies to non-square matrices, but you’d need a calculator that accepts m x n dimensions.
What is the relationship between rank and nullity?
The Rank-Nullity Theorem states: Rank(A) + Nullity(A) = number of columns of A.
Why is the nullspace important?
It provides information about the solutions to Ax=0, the linear independence of columns, and properties of the linear transformation represented by A (e.g., injectivity).
What does it mean if a vector is in the nullspace?
If a vector x is in the nullspace of A, it means that the linear transformation represented by A maps x to the zero vector.

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