Nullspace Calculator
Calculate Nullspace (Kernel)
Enter the dimensions of your matrix and then its elements to find the basis for its nullspace and its nullity.
What is a Nullspace Calculator?
A nullspace calculator is a tool used in linear algebra to find the nullspace (or kernel) of a given matrix. The nullspace of a matrix A consists of all vectors x that, when multiplied by A, result in the zero vector (Ax = 0). The nullspace calculator automates the process of finding a basis for this nullspace and determining its dimension, known as the nullity.
This calculator is useful for students learning linear algebra, engineers, scientists, and anyone working with systems of linear equations or matrix transformations. It helps visualize and understand the set of solutions to homogeneous linear systems.
A common misconception is that the nullspace is just the zero vector. While the zero vector is always in the nullspace, the nullspace often contains infinitely many other vectors, forming a vector subspace.
Nullspace Formula and Mathematical Explanation
For an m x n matrix A, its nullspace, denoted N(A), is defined as:
N(A) = {x ∈ Rn | Ax = 0}
To find the nullspace, we solve the homogeneous system of linear equations Ax = 0. This is typically done by performing Gaussian elimination to transform matrix A into its Row Reduced Echelon Form (RREF). Once in RREF, we identify the pivot variables (corresponding to columns with leading 1s) and free variables (corresponding to columns without leading 1s).
The free variables can be set as parameters, and the pivot variables are expressed in terms of these parameters. By setting each free variable to 1 (and others to 0) in turn, we can find a set of linearly independent vectors that span the nullspace. This set of vectors forms a basis for the nullspace.
The dimension of the nullspace is called the nullity of A, and it equals the number of free variables. The Rank-Nullity Theorem states that for an m x n matrix A, rank(A) + nullity(A) = n (number of columns).
Variables Table
| Variable | Meaning | Unit/Type | Typical Range |
|---|---|---|---|
| A | The m x n matrix | Matrix | Real numbers |
| x | A vector in Rn | Vector | Real numbers |
| 0 | The zero vector in Rm | Vector | Zeros |
| m | Number of rows in A | Integer | 1 to 10 (in this calculator) |
| n | Number of columns in A | Integer | 1 to 10 (in this calculator) |
| rank(A) | Rank of matrix A (number of pivots) | Integer | 0 to min(m, n) |
| nullity(A) | Dimension of the nullspace of A (number of free variables) | Integer | 0 to n |
Practical Examples (Real-World Use Cases)
Example 1: Finding the Nullspace of a 2×3 Matrix
Consider the matrix A:
A = | 1 2 3 |
| 2 4 6 |
We want to find vectors x = [x1, x2, x3]T such that Ax = 0. The system is:
1*x1 + 2*x2 + 3*x3 = 0
2*x1 + 4*x2 + 6*x3 = 0
The RREF of A is:
RREF(A) = | 1 2 3 |
| 0 0 0 |
x1 is the pivot variable, x2 and x3 are free variables. From the first row: x1 + 2*x2 + 3*x3 = 0 => x1 = -2*x2 – 3*x3.
If x2=1, x3=0, then x1=-2. Vector v1 = [-2, 1, 0]T.
If x2=0, x3=1, then x1=-3. Vector v2 = [-3, 0, 1]T.
A basis for the nullspace is {[-2, 1, 0]T, [-3, 0, 1]T}, and the nullity is 2. The nullspace calculator would output these basis vectors.
Example 2: A Matrix with Zero Nullity
Consider the matrix B:
B = | 1 0 |
| 0 1 |
The RREF is the matrix itself. Both x1 and x2 are pivot variables, there are no free variables. The only solution to Bx = 0 is x1=0, x2=0, so x = [0, 0]T. The nullspace contains only the zero vector, the basis is the empty set (or {0}), and the nullity is 0. Our nullspace calculator would indicate a nullity of 0 and a basis consisting only of the zero vector if forced to give one, or state the basis is {}.
How to Use This Nullspace Calculator
- Enter Matrix Dimensions: Start by inputting the number of rows (m) and columns (n) of your matrix A into the respective fields. The calculator will dynamically create input fields for the matrix elements.
- Input Matrix Elements: Fill in the elements of your matrix A into the generated grid. Ensure you enter numerical values.
- Calculate: Click the “Calculate Nullspace” button.
- View Results: The calculator will display:
- The basis vectors for the nullspace of A.
- The nullity of A (dimension of the nullspace).
- The original matrix A and its RREF in a table.
- A chart showing the rank and nullity.
- Interpret: The basis vectors span the nullspace. Any linear combination of these vectors is a solution to Ax=0. The nullity tells you how many linearly independent solutions exist. Check out our linear algebra tools for more.
Key Factors That Affect Nullspace Results
- Matrix Dimensions (m and n): The number of rows and columns determines the size of the matrix and the maximum possible rank and nullity.
- Rank of the Matrix: The rank (number of pivot columns/linearly independent rows or columns) directly influences the nullity through the Rank-Nullity Theorem (rank + nullity = n). A higher rank means a lower nullity.
- Linear Dependence of Rows/Columns: If rows or columns are linearly dependent, it reduces the rank and increases the nullity, meaning a larger nullspace.
- Values of Matrix Elements: The specific numbers within the matrix dictate the relationships between rows and columns, leading to the RREF and thus the nullspace.
- Homogeneous System: The nullspace is specifically for the Ax=0 system. Non-homogeneous systems (Ax=b, b≠0) have solution sets related to the nullspace but are not the nullspace itself.
- Invertibility (for square matrices): An n x n matrix is invertible if and only if its rank is n, which means its nullity is 0. A non-invertible square matrix has a non-trivial nullspace (nullity > 0). You can check invertibility with a matrix operations calculator.
Frequently Asked Questions (FAQ)
- What is the difference between nullspace and kernel?
- The terms “nullspace” and “kernel” are synonymous when referring to a matrix A. Both describe the set of vectors x such that Ax = 0.
- What does it mean if the nullity is 0?
- If the nullity of a matrix A is 0, it means the nullspace contains only the zero vector. The only solution to Ax = 0 is x = 0. For a square matrix, this implies the matrix is invertible.
- How is the nullspace related to the column space and row space?
- The nullspace is orthogonal to the row space of the matrix A. The column space’s dimension is the rank, and the nullspace’s dimension is the nullity.
- Can a nullspace calculator handle any matrix size?
- Our online nullspace calculator is typically limited to a maximum size (e.g., 10×10) for practical performance. Larger matrices require more computational power.
- What if my matrix has complex numbers?
- This specific nullspace calculator is designed for matrices with real number entries. Calculating the nullspace for matrices with complex entries follows similar principles but requires complex arithmetic.
- What is the Rank-Nullity Theorem?
- The Rank-Nullity Theorem states that for any m x n matrix A, the rank of A plus the nullity of A equals n (the number of columns of A). rank(A) + nullity(A) = n. Our rank calculator can help find the rank.
- How do I find a basis for the nullspace?
- You find the RREF of the matrix, identify free and pivot variables, express pivot variables in terms of free variables, and then create vectors by setting one free variable to 1 and others to 0 at a time. The nullspace calculator does this automatically.
- Does the nullspace always form a subspace?
- Yes, the nullspace of an m x n matrix is always a subspace of Rn.
Related Tools and Internal Resources
- RREF Calculator: Find the Row Reduced Echelon Form of a matrix, a key step used by the nullspace calculator.
- Matrix Rank Calculator: Calculate the rank of a matrix, related to nullity by the Rank-Nullity theorem.
- Matrix Operations Calculator: Perform various operations like addition, subtraction, and multiplication of matrices.
- Eigenvalue and Eigenvector Calculator: Find eigenvalues and eigenvectors, another important concept in linear algebra.
- Linear Algebra Tools: A collection of calculators for various linear algebra problems.
- Vector Calculator: Perform operations on vectors, which form the nullspace.