Answer to Question #19322 in Differential Geometry for ran

Question

if the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^ip_i$ where $p_i$ is a given covariant vector ,then $a_ij+a_ji=p_iv_j+p_jv_i$ where $v_j$ is some covariant vector