2.4 Composition of linear transformations

Linear transformations are powerful tools for manipulating vectors and spaces. Composition lets us combine these transformations, creating more complex operations from simpler ones. This idea is key to understanding how multiple transformations work together.

Composing transformations isn't just about math - it has real-world applications too. In computer graphics, game design, and physics, we use composition to create intricate movements and effects. It's a fundamental concept that bridges theory and practice in linear algebra.

Composition of Linear Transformations

Definition and Properties

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• Composition of linear transformations applies one linear transformation after another, resulting in a new linear transformation
• Denoted as S ∘ T, where ∘ represents the composition operator
• (S ∘ T)(v) equates to S(T(v)), meaning T applies first, followed by S
• Domain of S ∘ T matches the domain of T, while the codomain matches the codomain of S
• Preserves linearity, ensuring the resulting transformation remains linear
• Order of composition matters (S ∘ T ≠ T ∘ S in general)
  • Example: Rotating 90° clockwise then translating 2 units right differs from translating 2 units right then rotating 90° clockwise
  • Example: Scaling by factor 2 then reflecting over y-axis differs from reflecting over y-axis then scaling by factor 2

Geometric Interpretation

• Combines effects of individual transformations into a single operation
• Allows complex transformations by sequencing simpler ones
• Useful for analyzing compound movements in physics and computer graphics
  • Example: In 2D graphics, compose rotation and scaling to create a spiral effect
  • Example: In 3D modeling, combine translation, rotation, and scaling to position and orient objects

Computing Linear Transformation Compositions

Step-by-Step Computation

• Apply T to a general vector v, then apply S to the result
• For matrix representations, multiply corresponding matrices in reverse order of composition
• When composing multiple transformations, work from right to left: (R ∘ S ∘ T)(v) = R(S(T(v)))
• Verify compatibility of vector space dimensions in each composition step
• Express the resulting transformation as a single matrix or function based on context
  • Example: Compose rotation by 45° and scaling by factor 2 in 2D
  • Example: Combine reflection over x-axis and translation by (3, 4) in 2D

Practical Applications

• Practice composing common transformations in 2D and 3D spaces (rotations, reflections, scaling)
• Analyze geometric interpretations of composed transformations to understand combined effects
• Apply composition to solve problems in linear algebra and geometry
  • Example: Determine the single transformation equivalent to rotating 30° then reflecting over y-axis
  • Example: Find the matrix representing a 90° rotation followed by a doubling in size in 3D space

Associativity of Composition

Properties and Proofs

• Composition of linear transformations exhibits associativity: (R ∘ S) ∘ T = R ∘ (S ∘ T)
• Allows flexible grouping of transformations without altering the final result
• Prove associativity using composition definition and linear transformation properties
• Relates to associativity of matrix multiplication
• Associativity does not imply commutativity; transformation order remains crucial
  • Example: ((Rotation 45°) ∘ (Scale 2)) ∘ (Translate (1,0)) = (Rotation 45°) ∘ ((Scale 2) ∘ (Translate (1,0)))
  • Example: (Reflect y-axis) ∘ ((Rotate 90°) ∘ (Scale 3)) = ((Reflect y-axis) ∘ (Rotate 90°)) ∘ (Scale 3)

Applications and Implications

• Apply associativity to simplify complex compositions and prove related theorems
• Explore associativity's role in optimizing transformation calculations for computer graphics
• Understand how associativity affects the implementation of transformation sequences in programming
  • Example: Optimize a sequence of 3D transformations by grouping rotations together
  • Example: Use associativity to rearrange matrix multiplications for improved computational efficiency

Composition vs Matrix Multiplication

Relationship and Representation

• S ∘ T represented by matrix product BA, where A and B represent T and S respectively
• Matrix multiplication order reverses composition order: if S ∘ T = R, then BA = C (C represents R)
• Prove matrix multiplication correctly represents composition using multiplication definition and transformation properties
• Understand how matrix dimensions relate to vector space dimensions in transformations
  • Example: 2x2 matrix multiplied by 2x3 matrix represents composition of transformation from 3D to 2D followed by transformation within 2D
  • Example: 3x3 matrix multiplied by 3x2 matrix represents composition of transformation from 2D to 3D followed by transformation within 3D

Properties and Applications

• Explore how matrix multiplication properties reflect composition properties (non-commutativity, associativity)
• Use matrix multiplication for efficient computation of multiple linear transformation compositions
• Analyze how resulting matrix entries relate to geometric effects of composed transformations
• Apply matrix composition to solve problems in linear algebra and related fields
  • Example: Determine the matrix representing a sequence of 3D rotations about different axes
  • Example: Use matrix composition to find the inverse of a compound transformation