On the determinant
Let \(K\) be a skew field, let \(V\) denote a finite-dimensional left vector space over \(K\), and let \(GL(V)\) be the group of invertible linear operators on \(V\). A mapping \(\pi\in GL(V)\) is called perspective if there exist \(v\in V\) and \(\phi\in V^*\) (the right dual space of \(V\)) such that \((v, \phi)\neq 1\) and \(x\pi= x- (x, \phi)v\) for any \(x\in V\). It is well known that each element of \(GL(V)\) can be represented as a product of nontrivial perspective mappings (i.e. ``shears'' and ``dilatations'') \((*)\). The purpse of this note is to give a coordinate free definition of the determinant by using \((*)\). If \(P\) denotes the set of perspective mappings, then the mapping \(\text{det}: P\to K^*/(K^*)'\) (where \(K^*\) is the multiplicative group of \(K\) and \((K^*)'\) is the commutator group of \(K\)) is defined by \[ \text{det }\pi= \begin{cases} 1 & \text{whenever}\quad \pi= 1,\\ 1- (v, \phi) & \text{whenever}\quad \pi= \pi(v, \phi)\neq 1.\end{cases}\tag{\(**\)} \] The author extends this notion in the following way: if \(\sigma\in GL(V)\), then, by \((*)\), there exist perspective mappings \(\pi_1,\dots, \pi_n\) such that \((***)\) \(\sigma= \pi_1\cdots \pi_n\). Now, introduce the mapping \(\text{det}: GL(V)\to K^*/(K^*)'\) by \(\text{det }\sigma := \text{det }\pi_1\cdots \text{det } \pi_n\), where the factors \(\text{det } \pi_i\) are given by \((**)\). This definition of the determinant det is independent of the special representation \((***)\).
scientific article; zbMATH DE number 495263