The items starting this question appeared as Exercise 32.
(a) Prove that the composition of the projections πx, πy: R3 → R3 is the zero map despite that neither is the zero map.
(b) Prove that the composition of the derivatives d2/dx2, d3/dx3: P4 → P4 is the zero map despite that neither map is the zero map.
(c) Give matrix equations representing each of the prior two items. When two things multiply to give zero despite that neither is zero, each is said to be a zero divisor. Prove that no zero divisor is invertible.