Show that the composition of two linear maps is linear.

Question

Let T ∈ Linear map: U → V.

Let S ∈ Linear map: V → W.

Show that the composition ST: U → W is linear.

Answer 1

Let u be be an arbitrary vector in U and a be an element of some field, e.g. the complex numbers. Note that Tu is a vector in V. Then STau=S(Tau)=SaTu = aSTu. This is because S and T are individually linear. Thus, ST commutes with scalar multiplication. Then, if u, w are arbitrary vectors in U, ST(u+w)=S(Tu+Tw)=STu+STw, again because S and T are individually linear. These are the two properties needed for a linear map: that ST preserves scalar multiplication and addition.