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 \(W\).
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