math111_logo Composition and Matrix Multiplication

Extra: Composition of linear transformations is linear

Let T: RmRn and S: RkRn be linear transformations. Then by the definition, we have

  1. T(u + v) = T(u) + T(v),
  2. T(cu) = cT(u),
  3. S(u + v) = S(u) + S(v),
  4. S(cu) = cS(u).

To show the composition TS: RkRm is also linear means the verification of the following equalities:

TS(u + v) = TS(u) + TS(v),
TS(cu) = cTS(u).

The first equality is verified as follows:

TS(u + v) = T(S(u + v)) (definition of composition)
= T(S(u) + S(v)) (property 3. above)
= T(S(u)) + T(S(v)) (property 1. above)
= TS(u) + TS(v) (definition of composition)

The second equality is verified as follows:

TS(cu) = T(S(cu)) (definition of composition)
= T(Cs(u)) (property 4. above)
= cT(S(u)) (property 2. above)
= CTS(u) (definition of composition)

This completes the proof that TS is a linear transformation.