(a) Show that composition of linear functions is associative: (L M) N = L ...

Question:

(a) Show that composition of linear functions is associative: (L ○ M) ○ N = L ○ (M ○ N). Be precise about the domain and target spaces involved.
(b) How do you know the result is a linear function?
(c) Explain why this proves associativity of matrix multiplication.