67. Null Space

The null space (or kernel) of a matrix is the set of all vectors that satisfy:

For an matrix :

The null space is a subspace of . By the Rank–Nullity Theorem:

67.1. Computing the null space

Row-reduce to RREF. The non-pivot columns correspond to free variables; the pivot columns are expressed in terms of those free variables. A basis for the null space is built from one vector per free variable.

Example

Find with :

Augmented matrix:

Row-reduce to find the null space. (See Gaussian Elimination.)

67.2. Nullity

The nullity of is the dimension of its null space:

After row-reducing to RREF, nullity equals the number of non-pivot columns (= number of free variables).

67.3. Kernel vs null space

For a linear transformation :

Same set, different names: kernel is the linear-map perspective, null space is the matrix perspective. See Kernel.

67.4. Left null space

The null space of is called the left null space of :

It’s orthogonal to the column space of . Together with and , these are the four fundamental subspaces.

67.5. See also