Linear Algebra
4.110 min read

Composition of Linear Maps

When we apply two linear maps in sequence — first BB, then AA — the result is their composition (A∘B)(v)=A(Bv)(A \circ B)(\mathbf{v}) = A(B\mathbf{v}). The composition of two linear maps is itself a linear map.

This is the conceptual origin of matrix multiplication. If B:Rp→RnB: \mathbb{R}^p \to \mathbb{R}^n and A:Rn→RmA: \mathbb{R}^n \to \mathbb{R}^m, then A∘B:Rp→RmA \circ B: \mathbb{R}^p \to \mathbb{R}^m. The inner dimensions nn must match — this is exactly the dimension-matching rule for matrix multiplication.

Why study composition? Because in practice, transformations are always chained: rotate then scale, project then filter, encode then decode. Each step is a matrix; chaining steps is matrix multiplication.

Formal View

Definition 4.1 — Composition
If B:Rp→RnB: \mathbb{R}^p \to \mathbb{R}^n and A:Rn→RmA: \mathbb{R}^n \to \mathbb{R}^m are linear maps, their composition is (A∘B):Rp→Rm(A \circ B): \mathbb{R}^p \to \mathbb{R}^m defined by (A∘B)(v)=A(Bv)(A \circ B)(\mathbf{v}) = A(B\mathbf{v}). This map is also linear.

Interactive Visualization

Composition: v → Bv → A(Bv)

Why This Matters

Composition is the mathematical model of "apply transformation A after transformation B."

  • Computer graphics: model → world → camera → clip → screen space are 4 successive matrix multiplications
  • Neural networks: each layer applies a linear map, and forward pass is a composition of all layers
  • Signal processing: cascaded filters correspond to composed linear maps

Quiz

Question 1

The composition of two linear maps is always a linear map.

Common Mistakes

  • Thinking (A∘B)v=Av(A \circ B)\mathbf{v} = A\mathbf{v} then BB of the result — composition applies BB FIRST, then AA.
  • Ignoring dimension compatibility — BB must output vectors that AA can accept as input.