The Role of Determinants

The determinant is one of the most exciting and essential functions in mathematics and physics. Its significance stems from the fact that it is a profoundly geometric object. It possesses many manifestations. Its domain is usually the \(n \times n\) -matrices, and it may also be called the determinant function. Another form is the map from the Cartesian product \(V^n = V \times \dotsc \times V\) to \(\mathbb K\) (where \(\dim V = n\) ) which is linear in every component with the additional property (alternating) that if two vectors are identical, the result is zero. This is usually called a multilinear alternating form or a determinant form or even a volume form on V. In connection with this, the notion of orientation is illuminated by a determinant. In what follows, we start with the algebraic point of view for determinants, and in doing so, we derive and discuss most of the properties of determinants. Later we address the geometric point of view. In addition, we define the determinant of a linear operator, which is essentially a third manifestation of determinants.