Find Nullspace Of Matrix Calculator

Easily find the basis for the nullspace and the nullity of any matrix.

Matrix Input

What is the Nullspace of a Matrix?

The nullspace (or kernel) of a matrix A is the set of all vectors x that satisfy the linear equation Ax = 0 (where 0 is the zero vector). It is a vector subspace of the domain of the linear transformation represented by matrix A. In simpler terms, it’s the collection of all vectors that, when multiplied by the matrix A, result in the zero vector.

The nullspace is fundamental in linear algebra and has applications in various fields, including computer graphics, engineering, and data analysis. Understanding the nullspace helps us understand the solutions to systems of linear equations and the properties of linear transformations.

Anyone working with linear algebra, systems of equations, or transformations, such as students, engineers, data scientists, and mathematicians, should use and understand the nullspace. Common misconceptions include thinking the nullspace always contains only the zero vector (it does, but often more) or that it’s the same as the column space.

Our Nullspace of a Matrix Calculator helps you find a basis for this subspace and its dimension (nullity).

Nullspace Formula and Mathematical Explanation

To find the nullspace of a matrix A, we solve the homogeneous system of linear equations Ax = 0. The process usually involves these steps:

1. Row Reduction: Transform the matrix A into its Row Reduced Echelon Form (RREF) using Gaussian elimination or Gauss-Jordan elimination. Let’s call the RREF matrix R.
2. Identify Pivot and Free Variables: In the RREF matrix R, identify the pivot columns (columns containing leading 1s) and the free columns (columns without leading 1s). The variables corresponding to pivot columns are basic (or pivot) variables, and those corresponding to free columns are free variables.
3. Express Basic Variables: Write the equations from Rx = 0 and express each basic variable in terms of the free variables.
4. Form Basis Vectors: For each free variable, create a vector by setting that free variable to 1 and all other free variables to 0, then solve for the basic variables. The collection of these vectors forms a basis for the nullspace of A. The number of these vectors is the nullity of A.

The dimension of the nullspace is called the nullity of the matrix, which is equal to the number of free variables (or columns without pivots in the RREF).

Variables Table

Variable	Meaning	Unit	Typical Range
A	The input matrix	Matrix elements	Real numbers
x	Vector in the domain	Vector elements	Real numbers
R	Row Reduced Echelon Form of A	Matrix elements	Real numbers (often 0s and 1s)
Nullity(A)	Dimension of the nullspace of A	Integer	0 to n (number of columns)

Our Nullspace of a Matrix Calculator performs these steps to give you the basis vectors and the nullity.

Practical Examples (Real-World Use Cases)

Example 1: A 2×3 Matrix

Consider the matrix A = [[1, 2, 3], [2, 4, 6]]. We want to find its nullspace.

1. The system Ax = 0 is:
x + 2y + 3z = 0
2x + 4y + 6z = 0

2. RREF of A: [[1, 2, 3], [0, 0, 0]]. Pivot is in the first column (x). y and z are free variables.

3. From RREF: x + 2y + 3z = 0 => x = -2y – 3z

4. Basis vectors:
Let y=1, z=0 => x=-2. Vector: [-2, 1, 0]^T
Let y=0, z=1 => x=-3. Vector: [-3, 0, 1]^T

The nullspace is spanned by {[-2, 1, 0]^T, [-3, 0, 1]^T}, and the nullity is 2. The Nullspace of a Matrix Calculator would output these basis vectors.

Example 2: A 3×3 Matrix

Consider the matrix B = [[1, 0, 1], [0, 1, 1], [0, 0, 0]]. This is already in RREF.

1. Pivots in columns 1 and 2 (x, y). z is free.

2. x + z = 0 => x = -z
y + z = 0 => y = -z

3. Let z=1 => x=-1, y=-1. Basis vector: [-1, -1, 1]^T

The nullspace is spanned by {[-1, -1, 1]^T}, nullity is 1. Check this with the Nullspace of a Matrix Calculator.

You might find related tools like our RREF calculator useful.

How to Use This Nullspace of a Matrix Calculator

1. Enter Dimensions: Input the number of rows and columns for your matrix. The calculator will dynamically create input fields for the matrix elements.
2. Enter Matrix Elements: Fill in the values for each element of your matrix A.
3. Calculate: Click the “Calculate Nullspace” button.
4. View Results: The calculator will display:
   • The Row Reduced Echelon Form (RREF) of your matrix.
   • The basis vectors for the nullspace. If the nullspace only contains the zero vector, it will indicate that.
   • The nullity of the matrix (the dimension of the nullspace).
5. Interpret: The basis vectors span the nullspace. Any linear combination of these vectors is a solution to Ax = 0.

Key Factors That Affect Nullspace Results

The nullspace and its basis are determined solely by the elements of the matrix A.

1. Linear Dependence of Rows/Columns: If rows or columns are linearly dependent, the matrix will have free variables after row reduction, leading to a non-trivial nullspace (nullity > 0).
2. Rank of the Matrix: The rank of a matrix (number of pivots in RREF) and its nullity are related by the Rank-Nullity Theorem: rank(A) + nullity(A) = number of columns of A. A lower rank means a higher nullity.
3. Matrix Dimensions: If a matrix has more columns than rows, it is guaranteed to have a non-trivial nullspace (nullity >= number of columns – number of rows).
4. Invertibility: For a square matrix, it is invertible if and only if its nullspace is trivial (contains only the zero vector, nullity = 0).
5. Zero Rows in RREF: The presence of zero rows in the RREF indicates linear dependence and contributes to free variables and a non-trivial nullspace.
6. Exact Values of Elements: Small changes in matrix elements can drastically change the RREF and thus the nullspace, especially if they alter the linear dependence relationships. Consider using an eigenvalue calculator for related matrix properties.

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 such that Ax = 0.

What is the nullity of a matrix?

The nullity is the dimension of the nullspace, which is equal to the number of free variables in the RREF of the matrix.

How do I find the nullspace using the Nullspace of a Matrix Calculator?

Enter the matrix dimensions and elements, then click “Calculate Nullspace”. The calculator will show the RREF, basis vectors, and nullity.

What if the nullspace only contains the zero vector?

The calculator will indicate that the basis is empty or consists only of the zero vector, and the nullity will be 0. This happens for invertible square matrices.

Can the Nullspace of a Matrix Calculator handle non-square matrices?

Yes, it can calculate the nullspace for any m x n matrix.

What is the relationship between rank and nullity?

The Rank-Nullity Theorem states: rank(A) + nullity(A) = n (number of columns of A). You might also be interested in our matrix inverse calculator for square matrices.

Why is the nullspace important?

It helps solve homogeneous systems of equations, understand linear transformations (what vectors map to zero), and determine linear independence.

Is the nullspace always a vector space?

Yes, the nullspace of any matrix is always a subspace of Rⁿ (where n is the number of columns).

Related Tools and Internal Resources

• RREF Calculator: Find the Row Reduced Echelon Form of a matrix.
• Eigenvalue and Eigenvector Calculator: Calculate eigenvalues and eigenvectors for square matrices.
• Matrix Inverse Calculator: Find the inverse of a square matrix.
• Determinant Calculator: Calculate the determinant of a square matrix.
• Linear Independence Checker: Check if a set of vectors is linearly independent.
• Vector Span Calculator: Find the span of a set of vectors.