Definition of onto/surjective linear transformation

Definition: A linear transformation (T: V \to W) is onto (or surjective) if every vector in the codomain (W) is the image of at least one vector in the domain (V).

Equivalently: (T) is onto iff (\text{range}(T) = W).

Equivalently: For every (\mathbf{w} \in W), there exists (\mathbf{v} \in V) such that (T(\mathbf{v}) = \mathbf{w}).

Example (onto): The projection (T: \mathbb{R}^3 \to \mathbb{R}^2) defined by (T(x,y,z) = (x,y)) is onto because every vector ((a,b) \in \mathbb{R}^2) is the image of ((a,b,0)).

Example (not onto): The map (T: \mathbb{R}^2 \to \mathbb{R}^3) defined by (T(x,y) = (x,y,0)) is not onto because no vector maps to ((0,0,1)).

Key fact: If (T) is represented by matrix (A), then (T) is onto iff the columns of (A) span (W).