Label the following statements as true or false. (a) Every quadratic form is a bilinear

Chapter 6, Problem 1

(choose chapter or problem)

Label the following statements as true or false. (a) Every quadratic form is a bilinear form. (b) If two matrices are congruent, they have the same eigenvalues. (c) Symmetric bilinear forms have symmetric matrix representations. (d) Any symmetric matrix is congruent to a diagonal matrix. (e) The sum of two symmetric bilinear forms is a symmetric bilinear form. (f) Two symmetric matrices with the same characteristic polynomial are matrix representations of the same bilinear form. (g) There exists a bilinear form 77 such that H(x,y) ^ 0 for all x and y(h) If V is a vector space of dimension n, then dim(B(V)) = 2n. (i) Let 77 be a bilinear form on a finite-dimensional vector space V with dim(V) > 1. For any x G V, there exists y G V such that y^0,butH(x,y) = 0. (j) If 77 is any bilinear form on a finite-dimensional real inner product space V, then there exists an ordered basis 0 for V such that ijjp(H) is a diagonal matrix.