Null Space

67. Null Space

The null space (or kernel) of a matrix $𝑨$ is the set of all vectors $\vec{𝑥}$ that satisfy:

𝑨 \vec{𝑥} = 𝟎

For an $𝑚 \times 𝑛$ matrix $𝐴$ :

$\vec{𝑥} \in ℝ^{𝑛}$
$𝟎 \in ℝ^{𝑚}$

The null space is a subspace of $ℝ^{𝑛}$ . By the Rank–Nullity Theorem:

dim (Null (𝐴)) = 𝑛 - rank (𝐴)

67.1. Computing the null space

Row-reduce $𝐴$ 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

𝑨 = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{matrix}]

Find $\vec{𝑥} = [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ 𝑥_{3} \\ 𝑥_{4} \end{matrix}]$ with $𝑨 \vec{𝑥} = 𝟎$ :

{\begin{cases} 1 𝑥_{1} + 1 𝑥_{2} + 1 𝑥_{3} + 1 𝑥_{4} = 0 \\ 1 𝑥_{1} + 2 𝑥_{2} + 3 𝑥_{3} + 4 𝑥_{4} = 0 \\ 4 𝑥_{1} + 3 𝑥_{2} + 2 𝑥_{3} + 1 𝑥_{4} = 0 \end{cases}

Augmented matrix:

[\begin{matrix} 1 & 1 & 1 & 1 & 0 \\ 1 & 2 & 3 & 4 & 0 \\ 4 & 3 & 2 & 1 & 0 \end{matrix}]

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

67.2. Nullity

The nullity of $𝐴$ is the dimension of its null space:

nullity (𝐴) = dim (Null (𝐴))

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

67.3. Kernel vs null space

For a linear transformation $𝑇 (\vec{𝑥}) = 𝐴 \vec{𝑥}$ :

ker (𝑇) = Null (𝐴)

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 $𝐴^{𝑇}$ is called the left null space of $𝐴$ :

Null (𝐴^{𝑇}) = {\vec{𝑦} : {\vec{𝑦}}^{𝑇} 𝐴 = 𝟎}

It’s orthogonal to the column space of $𝐴$ . Together with $Null (𝐴)$ and $Row (𝐴)$ , these are the four fundamental subspaces.

67.5. See also

Kernel — same concept, transformation perspective
Rank–Nullity Theorem
Rank
Homogeneous System
Column Space — the “output side” counterpart