What is the Null Space of a Matrix?
In linear algebra, the null space (or kernel) of an m x n matrix A is a vector subspace of ℝn containing all vectors x that map to the zero vector under the linear transformation defined by A. In other words, it is the complete set of solutions to the homogeneous system of equations:
A × x = 0
To define the null space completely, we find its basisâ€â€a set of linearly independent vectors that span the entire subspace. Any vector in the null space can be written as a linear combination of these basis vectors.
How to Find the Null Space Basis Using RREF
The standard method to find the basis of the null space is to solve the homogeneous system using row reduction:
- Reduce the matrix A to its Reduced Row Echelon Form (RREF) using Gauss-Jordan elimination.
- Identify the pivot columns (columns that contain a leading 1) and the free columns (columns that do not contain a pivot).
- Write the system of equations corresponding to the RREF. Each row represents a pivot variable expressed in terms of the free variables.
- For each free variable, create a basis vector by setting that free variable to 1 and all other free variables to 0, and then solving for the pivot variables.
- The resulting vectors form the basis for the null space.
Rank-Nullity and Linear Independence
The null space is directly related to the concept of linear independence. If the null space contains *only* the zero vector (meaning there are no free variables in RREF), it indicates that the columns of A are **linearly independent**. In this case, the nullity of the matrix is 0.
According to the Rank-Nullity Theorem, the number of columns in A is equal to the rank of A (number of pivots) plus the nullity (number of free variables):
rank(A) + nullity(A) = n
Finding the null space is a critical step in solving non-homogeneous systems Ax = B, as the general solution is the sum of a particular solution and the null space solutions.