Chapter 3. Bilinear forms Lecture notes for MA1212 P. Karageorgis
[email protected] 1/25 Bilinear forms Definition 3.1 – Bilinear form A bilinear form on a real vector space V is a function f : V V R × → which assigns a number to each pair of elements of V in such a way that f is linear in each variable. A typical example of a bilinear form is the dot product on Rn. We shall usually write x, y instead of f(x, y) for simplicity and we h i shall also identify each 1 1 matrix with its unique entry. × Theorem 3.2 – Bilinear forms on Rn Every bilinear form on Rn has the form t x, y = x Ay = aijxiyj h i Xi,j for some n n matrix A and we also have aij = ei, ej for all i,j. × h i 2/25 Matrix of a bilinear form Definition 3.3 – Matrix of a bilinear form Suppose that , is a bilinear form on V and let v1, v2,..., vn be a h i basis of V . The matrix of the form with respect to this basis is the matrix A whose entries are given by aij = vi, vj for all i,j. h i Theorem 3.4 – Change of basis Suppose that , is a bilinear form on Rn and let A be its matrix with respect to theh standardi basis. Then the matrix of the form with respect t to some other basis v1, v2,..., vn is given by B AB, where B is the matrix whose columns are the vectors v1, v2,..., vn.