Exercise 23 — Linear Algebra

The infinite-dimensional space $\mathcal{P}$ of all finite-degree polynomials gives a memorable example of the non-commutativity of linear maps. Let $d/dx\colon\mathcal{P}\to\mathcal{P}$ be the usual derivative and let $s\colon\mathcal{P}\to\mathcal{P}$ be the shift map.

a_{0}+a_{1}x+\dots+a_{n}x^{n}\;\stackrel{{\scriptstyle s}}{{\longmapsto}}\;0+a_{0}x+a_{1}x^{2}+\dots+a_{n}x^{n+1}

Show that the two maps don’t commute $d/dx\mathbin{\circ}s\neq s\mathbin{\circ}d/dx$ ; in fact, not only is $(d/dx\mathbin{\circ}s)-(s\mathbin{\circ}d/dx)$ not the zero map, it is the identity map.

Hefferon, Linear Algebra, Maps Between Spaces, Matrix Multiplication, Exercise 23

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