Image of a linear map

Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a linear map. The image of $T$, also called the range, is $$\im T = \{T(v) : v \in V\} \subseteq W.$$

Remarks

The image is a linear subspace of $W$. For the map $x \mapsto Ax$ of a matrix $A$, the image is the column space of $A$. The map $T$ is surjective if and only if $\im T = W$. The dimension of the image is the rank of $T$.

Explanations

L1 human

The image of a linear map is everything that the map can produce. If a map sends ordinary space onto a plane, then its image is that plane. The image is always a subspace.