Determinant/R/Relation to volume/Remark

In case ${}K=\mathbb {R}$, the determinant is in tight relation to volumes of geometric objects. If we consider in ${}\mathbb {R} ^{n}$ vectors ${}v_{1},\ldots ,v_{n}$, then they span a parallelotope. This is defined by

${}P:={\left\{s_{1}v_{1}+\cdots +s_{n}v_{n}\mid s_{i}\in [0,1]\right\}}\,.$

It consists of all linear combinations of these vectors, where all the scalars belong to the unit interval. If the vectors are linearly independent, then this is a "voluminous“ body, otherwise it is an object of smaller dimension. Now the relation

${}\operatorname {vol} \,P=\vert {\det {\left(v_{1},\ldots ,v_{n}\right)}}\vert \,$

holds, saying that the volume of the parallelotope is the modulus of the determinant of the matrix, consisting of the spanning vectors as columns (or rows).