Nullspace Calculator
An SEO expert and frontend developer’s tool to compute the basis of the null space (kernel) for any matrix.
What is a Nullspace?
In linear algebra, the nullspace (or kernel) of a matrix A is the set of all vectors x that solve the homogeneous equation Ax = 0. In other words, it’s the collection of all vectors that, when multiplied by the matrix A, result in the zero vector. This concept is fundamental to understanding the properties of linear transformations. The nullspace is itself a vector space, and this nullspace calculator is designed to find a ‘basis’ for it – a set of linearly independent vectors that span the entire nullspace.
Nullspace Formula and Explanation
There isn’t a single “formula” for the nullspace, but rather a process to find its basis. The core of this process is solving the system of linear equations represented by Ax = 0. To do this efficiently, we use a method called Gaussian elimination to transform the matrix A into its Reduced Row Echelon Form (RREF).
Once the matrix is in RREF, we identify two types of variables:
- Pivot variables: Variables corresponding to columns that contain a leading ‘1’ (a pivot).
- Free variables: Variables corresponding to columns that do not have a pivot.
The solution is expressed by writing the pivot variables in terms of the free variables. For each free variable, we construct a basis vector by setting that free variable to 1 and all other free variables to 0, then solving for the pivot variables. The collection of these vectors forms the basis for the nullspace.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| A | The input matrix | Unitless | Any real numbers |
| x | The vector solution | Unitless | Any real numbers |
| 0 | The zero vector | Unitless | All entries are zero |
| Basis Vectors | A set of linearly independent vectors that span the nullspace | Unitless | Real-numbered vectors |
Practical Examples
Example 1: A 2×3 Matrix
Consider the matrix A:
[,]
Using the nullspace calculator, we would first find its RREF:
[[1, 0, -1],]
This corresponds to the equations x1 – x3 = 0 and x2 + 2*x3 = 0. Here, x3 is the free variable. Setting x3 = 1, we solve to get x1 = 1 and x2 = -2. Therefore, the basis for the nullspace is the single vector:
[, [-2],]
Example 2: A 3×3 Matrix with Zero Nullity
Consider the identity matrix A:
[,,]
This matrix is already in RREF. There are no free variables. The only solution to Ax = 0 is the trivial solution, x = 0. In this case, the nullspace contains only the zero vector, and its basis is the empty set. The dimension of the nullspace (the nullity) is 0.
How to Use This Nullspace Calculator
- Set Matrix Dimensions: Use the “Rows” and “Columns” input fields to define the size of your matrix. The grid will update automatically.
- Enter Matrix Values: Fill in each cell of the matrix grid with the appropriate numerical values. These values are treated as unitless real numbers.
- Calculate: Click the “Calculate Nullspace” button to perform the calculation.
- Interpret Results: The calculator will display two key pieces of information. First, the “Reduced Row Echelon Form (RREF)” of your matrix is shown as an intermediate step. Second, the “Basis for the Null Space” is presented, which is the primary result. This is a set of vectors that define the solution space for Ax=0.
- Trivial Nullspace: If the only solution is the zero vector, the calculator will indicate that the basis is the empty set and the nullity is 0.
Key Factors That Affect the Nullspace
- Matrix Rank: The rank of a matrix (the number of pivots in its RREF) is directly related to the nullity. The Rank-Nullity Theorem states that Rank(A) + Nullity(A) = Number of Columns. A higher rank means a lower nullity.
- Linear Dependence of Columns: If the columns of the matrix are linearly dependent, it means there are redundant vectors. This redundancy leads to the existence of free variables and thus a non-trivial nullspace.
- Matrix Dimensions: A “wide” matrix (more columns than rows) is guaranteed to have a non-trivial nullspace because there must be at least one free variable.
- Invertibility of the Matrix: For a square matrix, being invertible is equivalent to having a rank equal to its number of columns. This means the nullity is 0, and the nullspace contains only the zero vector. A non-invertible (singular) square matrix will have a non-trivial nullspace.
- Row Operations: Performing elementary row operations does not change the nullspace of a matrix. This is why Gaussian elimination is a valid method for finding it.
- Scalar Multiplication: Multiplying the entire matrix by a non-zero scalar does not change its nullspace.
Frequently Asked Questions (FAQ)
The nullspace is the set of vectors ‘x’ that solve Ax=0. The column space is the set of all possible linear combinations of the matrix’s column vectors. You can learn more about it with a Column Space Calculator.
A nullity of 0 means the dimension of the nullspace is zero. This indicates that the nullspace contains only the zero vector, and the columns of the matrix are linearly independent. The only solution to Ax=0 is the trivial solution x=0.
The nullspace is an abstract mathematical concept from linear algebra. The numbers in the matrix represent scalar quantities, not physical measurements with units like meters or kilograms. Therefore, the inputs and resulting basis vectors are unitless.
Kernel is another name for the nullspace. The terms are used interchangeably.
An eigenvector of a matrix A corresponding to an eigenvalue of 0 is a vector that lies in the nullspace of A. You can explore this further with an Eigenvalue Calculator.
This specific tool is optimized for matrices up to 10×10 for performance and usability reasons. The mathematical principle, however, applies to matrices of any size.
A set of vectors “spans” a space if every vector in that space can be written as a linear combination of the vectors in the set. The basis of a nullspace is a minimal set of vectors that spans that entire nullspace.
The Rank-Nullity Theorem is a fundamental theorem in linear algebra that states for a matrix A with ‘n’ columns, the rank of A plus the nullity of A equals ‘n’. This provides a powerful check on calculations involving these properties. A Matrix Rank Calculator can be a helpful companion tool.