Exercise 7 — Linear Algebra

Show that any permutation, any reordering, $p$ of the numbers $1$ , …, $n$ , the map

\begin{pmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{pmatrix}\mapsto\begin{pmatrix}x_{p(1)}\\ x_{p(2)}\\ \vdots\\ x_{p(n)}\end{pmatrix}

can be done with a composition of maps, each of which only swaps a single pair of coordinates. Hint: you can use induction on $n$ . (Remark: in the fourth chapter we will show this and we will also show that the parity of the number of swaps used is determined by $p$ . That is, although a particular permutation could be expressed in two different ways with two different numbers of swaps, either both ways use an even number of swaps, or both use an odd number.)

Hefferon, Linear Algebra, Maps Between Spaces, Geometry of Linear Maps, Exercise 7

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