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.
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