71. Composition of Linear Transformations

For two linear transformations and , the composition applies first, then :

The composition of two linear transformations is itself linear:

Additivity:

Homogeneity: same argument with scalar pulled through.

71.1. Composition ↔ matrix multiplication

If has matrix () and has matrix (), then:

So composition corresponds to matrix multiplication — with the matrices in reverse order (the transformation applied first sits on the right):

This single observation is the entire reason matrix multiplication is defined the way it is.

71.2. Properties of composition

• Associative: — follows from matrix multiplication being associative
• Not commutative: in general — different transformations don’t commute (and matrices generally don’t either)
• Identity: composing with (the identity transformation) leaves any transformation unchanged
• Inverse: (when defined) — peel off in reverse order

71.3. See also

• Linear Transformation
• Matrix Multiplication
• Matrix Representation — how transformations become matrices
• Matrix Inverse