Composition

Definition of Composition

Linear Transformation

Mapping T: V → W, where V, W are vector spaces over field F. Satisfies linearity: T(av + bw) = aT(v) + bT(w).

Composition Operation

Given T: U → V and S: V → W, composition S ∘ T: U → W defined by (S ∘ T)(u) = S(T(u)) for all u ∈ U.

Existence Criterion

Composition only defined if codomain of first map equals domain of second: codomain(T) = domain(S).

Summary

Composition forms new linear map by applying transformations sequentially; fundamental in constructing complex mappings.

Notation and Terminology

Standard Symbols

Composition denoted by "∘". If S, T are maps, S ∘ T means "apply T first, then S".

Alternative Notation

Sometimes written as ST to indicate matrix multiplication corresponding to composition.

Terminology

Terminology includes "first map", "second map", "composite map", "operator" when domain and codomain coincide.

Order of Application

Order important: S ∘ T ≠ T ∘ S in general; composition is not commutative.

Properties of Composition

Linearity

Composition of linear maps is linear: (S ∘ T)(a u + b v) = a (S ∘ T)(u) + b (S ∘ T)(v).

Associativity

(R ∘ S) ∘ T = R ∘ (S ∘ T) for all linear maps T, S, R with compatible domains/codomains.

Non-commutativity

S ∘ T generally ≠ T ∘ S; order of application affects result.

Identity Map

Existence of identity I: I ∘ T = T and T ∘ I = T for all T with matching domain or codomain.

Distributivity over Addition

S ∘ (T + U) = S ∘ T + S ∘ U; (S + R) ∘ T = S ∘ T + R ∘ T.

Matrix Representation

Matrix Correspondence

Linear map T: V→W corresponds to matrix M_T relative to chosen bases.

Composition as Matrix Product

For S: W→X and T: V→W with matrices M_S and M_T, matrix of S ∘ T is M_S M_T.

Order in Multiplication

Matrix multiplication order matches composition order: S ∘ T → M_S M_T, not M_T M_S.

Example Table

Associativity of Composition

Formal Statement

For T: U→V, S: V→W, R: W→X, (R ∘ S) ∘ T = R ∘ (S ∘ T).

Proof Sketch

Evaluate both sides at u ∈ U: ((R ∘ S) ∘ T)(u) = R(S(T(u))) = (R ∘ (S ∘ T))(u).

Implications

Allows unambiguous notation without parentheses for multiple compositions.

Matrix Analogy

Matrix multiplication associative: (M_R M_S) M_T = M_R (M_S M_T).

Domain and Codomain Considerations

Compatibility Condition

Composition defined only if codomain of first map = domain of second map.

Examples

T: R² → R³, S: R³ → R⁴ → S ∘ T defined; T: R² → R³, S: R² → R⁴ → composition undefined.

Subspace Mappings

Composition may restrict domain or codomain to subspaces for well-definedness.

Extension via Zero Maps

Undefined compositions can be extended trivially by zero maps, but lose meaningfulness.

Invertibility and Composition

Invertible Linear Maps

T invertible if ∃ T⁻¹: W → V with T⁻¹ ∘ T = I_V and T ∘ T⁻¹ = I_W.

Composition of Invertibles

If T and S invertible, then S ∘ T invertible with inverse T⁻¹ ∘ S⁻¹.

Non-invertible Compositions

Composition may lose invertibility if either factor is non-invertible.

Example

Let T: R² → R² be invertible,S: R² → R² invertible,Then (S ∘ T)⁻¹ = T⁻¹ ∘ S⁻¹. 

Examples of Compositions

Example 1: Scaling and Rotation

T scales vectors by 2, S rotates vectors by 90°. S ∘ T scales then rotates.

Example 2: Projection and Reflection

T projects R³ onto a plane, S reflects across a different plane; S ∘ T combines effects.

Example 3: Differentiation Operators

T = d/dx, S = d/dx on polynomial space; composition S ∘ T = d²/dx².

Example 4: Zero Map Composition

Any map composed with zero map yields zero map.

Composition in Operator Theory

Operators as Endomorphisms

Linear operators: linear maps from V to V. Composition defines operator multiplication.

Algebraic Structure

End(V) with composition is associative algebra over field F.

Spectral Implications

Composition affects eigenvalues and spectra; product of operators influences spectral radius.

Functional Calculus

Composition used in defining functions of operators, e.g. powers, exponentials.

Applications of Composition

Computer Graphics

Combining transformations: translation, rotation, scaling via matrix composition.

Control Theory

System dynamics represented as composition of state-space transformations.

Quantum Mechanics

Operators composed to represent sequences of quantum operations or observables.

Data Transformation Pipelines

Linear transformations composed to model pipeline stages in machine learning and signal processing.

Limitations and Counterexamples

Non-commutativity Limitations

Order sensitivity restricts interchangeability of transformations.

Domain Mismatch

Composition undefined if domain/codomain incompatible; limits chaining of arbitrary maps.

Loss of Properties

Composition may not preserve injectivity, surjectivity, or invertibility.

Example Counterexample

Let T: R² → R² be projection onto x-axis,S: R² → R² be projection onto y-axis,Then S ∘ T = zero map,T ∘ S ≠ zero map,Showing non-commutativity and information loss. 

Computational Aspects

Algorithmic Efficiency

Matrix multiplication complexity dominates; standard O(n³), optimized algorithms exist.

Numerical Stability

Composition can amplify rounding errors; conditioning important in numerical linear algebra.

Software Implementations

Libraries (e.g., LAPACK, Eigen) provide optimized functions for composing linear maps.

Sparse Matrices

Composition efficient if matrices are sparse; storage and multiplication optimized.

Parallelization

Matrix multiplication and thus composition parallelizable for large-scale computations.

References

• Axler, S., Linear Algebra Done Right, Springer, 3rd Edition, 2015, pp. 45–67.
• Halmos, P.R., Finite-Dimensional Vector Spaces, Springer, 1958, pp. 89–110.
• Lang, S., Linear Algebra, Springer, 3rd Edition, 1987, pp. 120–140.
• Horn, R.A., Johnson, C.R., Matrix Analysis, Cambridge University Press, 2012, pp. 205–230.
• Axler, S., Operator Theory, American Mathematical Society, 2001, pp. 54–78.