Definition of one-to-one/injective linear transformation
Created almost 9 years ago, updated 5 months ago
ID23845
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.
**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.