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The Null Space

The null space (or kernel) of $A$ is the set of all vectors that map to zero: $\text{Null}(A) = \{\mathbf{x} : A\mathbf{x} = \mathbf{0}\}$ . It is always a subspace of $\mathbb{R}^n$ (the domain).

The null space captures the "collapse" of the map — vectors in the null space all land at the same point (the origin). A large null space means the map loses a lot of information.

The nullity is the dimension of the null space: $\text{Nullity}(A) = \dim(\text{Null}(A))$ . Nullity equals the number of free variables when solving $A\mathbf{x} = \mathbf{0}$ .

Formal View

Definition 3.7 — Null Space and Nullity

The null space of

A \in \mathbb{R}^{m \times n}

\text{Null}(A) = \{\mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0}\}

This is always a subspace of

\mathbb{R}^n

. The nullity is

\text{Nullity}(A) = \dim(\text{Null}(A))

Why This Matters

The null space tells us exactly how much information is lost by applying the matrix.

A camera projection matrix has a nontrivial null space: depth information is lost
The null space of an audio filter gives the frequencies completely eliminated
In control theory, the null space of the output matrix gives unobservable state directions

Learning Resources

Null space and column space

3Blue1Brown

Visualizing the null space as the kernel of a transformation.

10 min

Quiz

Question 1

The null space of any matrix always contains the zero vector.

Common Mistakes

Confusing the null space (in the domain $\mathbb{R}^n$ ) with the column space (in the codomain $\mathbb{R}^m$ ).
Thinking the null space is "unimportant" — it determines whether solutions are unique.