Understanding the Null Space
The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. In other words, it consists of all solutions to the homogeneous system of linear equations Ax = 0. The null space is always a subspace of R^n, where n is the number of columns of A.
Key Concepts
Null Space Definition
All vectors x where Ax = 0.
Null(A) = {x : Ax = 0}
Rank-Nullity Theorem
Rank plus nullity equals the number of columns.
rank(A) + nullity(A) = n
Finding Null Space
Row reduce to RREF, then identify free variables.
RREF(A) -> free vars -> basis
How to Find the Null Space
- Form the augmented matrix: Write [A | 0] (though the zero column is usually omitted).
- Row reduce to RREF: Use Gaussian elimination to reduce A to reduced row echelon form.
- Identify pivot and free variables: Columns with leading 1s are pivot variables; others are free variables.
- Express solutions: Set each free variable to a parameter and express pivot variables in terms of free variables.
- Write basis vectors: Each free variable yields one basis vector for the null space.
Properties of Null Space
- The null space always contains the zero vector.
- If the null space contains only the zero vector, the matrix columns are linearly independent.
- The dimension of the null space (nullity) equals the number of free variables.
- For a square matrix, the null space is trivial (just {0}) if and only if the matrix is invertible.
Applications
The null space is fundamental in linear algebra and appears in solving differential equations, finding eigenvectors, understanding linear transformations, network analysis, data compression, and machine learning. It reveals the "redundancy" or "degrees of freedom" in a system of equations.