Range and Null Space

Definition

The range of a matrix $A \in \R ^{m \times n}$ is the set

\[ \Set{y \in \R ^m}{(\exists z \in \R ^n)(y = Az)}. \]

We often denote this set by $\Set{Az}{z \in \R ^n}$. It is a subset of $\R ^m$, the output space.

The nullspace of a matrix $A \in \R ^{m \times n}$ is the set

\[ \Set{z \in \R ^n}{Az = 0}. \]

It is a subset of $\R ^n$, the input space.