Exercise 23

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

• 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.