Definition of one-to-one/injective linear transformation

Definition: A linear transformation (T: V \to W) is one-to-one (or injective) if distinct vectors in the domain map to distinct vectors in the codomain.

Equivalently: If (T(\mathbf{u}) = T(\mathbf{v})), then (\mathbf{u} = \mathbf{v}).

For linear transformations, this is equivalent to: (\ker(T) = {\mathbf{0}}) (only the zero vector maps to zero).

Example (one-to-one): The rotation (T: \mathbb{R}^2 \to \mathbb{R}^2) by 90 degrees, (T(x,y) = (-y, x)), is one-to-one.

Example (not one-to-one): The projection (T: \mathbb{R}^3 \to \mathbb{R}^2) defined by (T(x,y,z) = (x,y)) is not one-to-one because (T(0,0,1) = T(0,0,2) = (0,0)).

Key fact: If (T) is represented by matrix (A), then (T) is one-to-one iff the columns of (A) are linearly independent.