Null Space Calculator – Matrix Kernel Finder
Find the null space and kernel of any matrix.
Table of Contents
How to Use
- Set the number of rows and columns for your matrix
- Enter the matrix values in each cell
- Click calculate to find the null space
- View the basis vectors, dimension, and RREF
What is the Null Space?
The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It's a subspace of the domain that gets mapped to the zero vector under the linear transformation represented by A.
The null space is denoted as N(A) or ker(A), and its dimension is called the nullity of the matrix.
How to Find the Null Space
To find the null space:
- Reduce the matrix to Row Reduced Echelon Form (RREF)
- Identify pivot columns (leading 1s) and free columns
- Express pivot variables in terms of free variables
- Write the general solution as a linear combination of basis vectors
Rank-Nullity Theorem
The Rank-Nullity Theorem states that for an m×n matrix A: rank(A) + nullity(A) = n, where n is the number of columns.
- Rank = number of pivot columns = dimension of column space
- Nullity = number of free variables = dimension of null space
- If nullity = 0, the only solution to Ax = 0 is x = 0
Applications
The null space has many applications:
- Solving homogeneous systems of linear equations
- Finding the general solution to Ax = b
- Determining linear independence of vectors
- Understanding linear transformations
- Computer graphics and data compression
Frequently Asked Questions
- What does a trivial null space mean?
- A trivial null space contains only the zero vector, meaning nullity = 0. This occurs when the matrix has full column rank, and the only solution to Ax = 0 is x = 0.
- How are free variables related to the null space?
- Free variables correspond to non-pivot columns in the RREF. Each free variable contributes one dimension to the null space, and the number of free variables equals the nullity.
- What's the difference between null space and column space?
- The null space is the set of vectors x where Ax = 0 (in the domain). The column space is the set of all possible outputs Ax (in the codomain). They are orthogonal complements in certain contexts.
- Can a square matrix have a non-trivial null space?
- Yes, if the square matrix is singular (determinant = 0). A non-trivial null space means the matrix is not invertible and has linearly dependent columns.