Linear Algebra
3.89 min read

Injectivity: No Collisions

A linear map A:Rn→RmA: \mathbb{R}^n \to \mathbb{R}^m is injective (or one-to-one) if different inputs always produce different outputs: x1≠x2⇒Ax1≠Ax2\mathbf{x}_1 \neq \mathbf{x}_2 \Rightarrow A\mathbf{x}_1 \neq A\mathbf{x}_2. No two inputs "collide" at the same output.

Equivalently (by linearity): Ax=0A\mathbf{x} = \mathbf{0} has only the trivial solution x=0\mathbf{x} = \mathbf{0}. The null space is trivial: Null(A)={0}\text{Null}(A) = \{\mathbf{0}\}.

Geometrically: the map is "faithful" — it doesn't "compress" any direction to zero. An injective map requires at least as many rows as columns: m≥nm \geq n.

Formal View

Definition 3.8 — Injectivity
A:Rn→RmA: \mathbb{R}^n \to \mathbb{R}^m is injective if Ax1=Ax2⇒x1=x2A\mathbf{x}_1 = A\mathbf{x}_2 \Rightarrow \mathbf{x}_1 = \mathbf{x}_2. Equivalently, Null(A)={0}\text{Null}(A) = \{\mathbf{0}\}, i.e., Nullity(A)=0\text{Nullity}(A) = 0.

Why This Matters

Injectivity means the map is reversible — from the output, we can uniquely recover the input.

  • A linear code is injective iff distinct messages produce distinct codewords
  • A linear measurement system is injective iff distinct signals produce distinct measurements
  • A transformation is injective iff it has a left inverse (can undo it)

Quiz

Question 1

A 2×32 \times 3 matrix can be injective?

Common Mistakes

  • Thinking injectivity requires the matrix to be square — it just requires m≥nm \geq n.
  • Confusing injective (no collisions) with surjective (covers everything).