71. Composition of Linear Transformations

For two linear transformations $𝑆:𝑋\to 𝑌$ and $𝑇:𝑌\to 𝑍$, the composition $𝑇\circ 𝑆:𝑋\to 𝑍$ applies $𝑆$ first, then $𝑇$:

The composition of two linear transformations is itself linear:

Additivity:

Homogeneity: same argument with scalar pulled through.

71.1. Composition ↔ matrix multiplication

If $𝑆$ has matrix $𝐴$ ($𝑚\times 𝑛$) and $𝑇$ has matrix $𝐵$ ($𝑙\times 𝑚$), 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: $\left(𝑅\circ 𝑇\right)\circ 𝑆=𝑅\circ \left(𝑇\circ 𝑆\right)$ — follows from matrix multiplication being associative
• Not commutative: in general $𝑇\circ 𝑆\ne 𝑆\circ 𝑇$ — different transformations don’t commute (and matrices generally don’t either)
• Identity: composing with $𝐼$ (the identity transformation) leaves any transformation unchanged
• Inverse: ${\left(𝑇\circ 𝑆\right)}^{-1}={𝑆}^{-1}\circ {𝑇}^{-1}$ (when defined) — peel off in reverse order

71.3. See also

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