Let m and n be positive integers and let F be an a field.
(a) Show that for every A E Mmn (F), there exists a unique linear mapping : Mn.1 (F)→ Mm.1 (F) such that [] A, where and are the standard ordered bases of Mn,1 (F) and Mm. 1 (F), respectively.
(b) Let: M₁,1 (F)→ Mm, 1 (F) be a linear mapping. Show that there exists a unique A E Mm,n (F) such that A. Here, 4 is the linear mapping from M₁1 (F) into Mm. 1 (F) given by A(u) = Au for every u € M₂, 1 (F) and is called the linear mapping associated to A.
(c) Let A E Mmn (F). Show that [VA] = A. where and are the standard ordered bases of Mn,1 (F) and Mm, 1 (F), respectively.