Bilinear Forms

The geometry of Rn is controlled algebraically by the dot product. We will abstract the dot product on Rn to a bilinear form on a vector space and study algebraic and geometric notions related to bilinear forms (especially the concept of orthogonality in all its manifestations: orthogonal vectors, orthogonal subspaces, and orthogonal bases). Section 1 defines a bilinear form on a vector space and offers examples of the two most common types of bilinear forms: symmetric and alternating bilinear forms. In Section 2 we will see how a bilinear form looks in coordinates. Section 3 describes the important condition of nondegeneracy for a bilinear form. Orthogonal bases for symmetric bilinear forms are the subject of Section 4. Symplectic bases for alternating bilinear forms are discussed in Section 5. Quadratic forms are in Section 6 (characteristic not 2) and Section 7 (characteristic 2). The tensor product viewpoint on bilinear forms is briefly discussed in Section 8. Vector spaces in Section 1 are arbitrary, but starting in Section 2 we will assume they are finite-dimensional. It is assumed that the reader is comfortable with abstract vector spaces and how to use bases of (finite-dimensional) vector spaces to turn elements of a vector space into column vectors and linear maps between vector spaces into matrices. It is also assumed that the reader is familiar with duality on finite-dimensional vector spaces: dual spaces, dual bases, the dual of a linear map, and the natural isomorphism of finite-dimensional vector spaces with their double duals (which identifies the double dual of a basis with itself and the double dual of a linear map with itself). For a vector space V we denote its dual space as V ∨. The dual basis of a basis {e1, . . . , en} of V is denoted {e1 , . . . , en}, so the ei ’s are the coordinate functions on V relative to that basis: ei (ej) is 1 for i = j and 0 for i 6= j. Although V is naturally isomorphic to V ∨∨, students are always cautioned against identifying V with V ∨, since “there is no natural isomorphism.” In a nutshell, the subject of bilinear forms is about what happens if we make an identification of V with V ∨ and keep track of it. Different identifications have different geometric properties.