Definition of onto/surjective linear transformation
Created almost 9 years ago, updated 5 months ago
ID23844
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).
**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 \(W\).