(We will use this exercise in the Matrix Inverses exercises.) Here is another property of matrix multiplication that might be puzzling at first sight.
(a) Prove that the composition of the projections πx, πy: R3 → R3 onto the x and y axes is the zero map despite that neither one is itself the zero map.
(b) Prove that the composition of the derivatives d2/dx2, d3/dx3: P4 → P4 is the zero map despite that neither is the zero map.
(c) Give a matrix equation representing the first fact.
(d) Give a matrix equation representing the second.
When two things multiply to give zero despite that neither is zero we say that each is a zero divisor.