Exercise 19

(We will use this exercise in the Matrix Inverses exercises.) Here is another property of matrix multiplication that might be puzzling at first sight.

• Prove that the composition of the projections $\pi_{x},\pi_{y}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ onto the $x$ and $y$ axes is the zero map despite that neither one is itself the zero map.

• Prove that the composition of the derivatives $d^{2}/dx^{2},\,d^{3}/dx^{3}\colon\mathcal{P}_{4}\to\mathcal{P}_{4}$ is the zero map despite that neither is the zero map.

• Give a matrix equation representing the first fact.

• 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.