Bilinear form/Symmetric/Definiteness/Minor criterion/Fact/Proof

Proof

(1). If the bilinear form is positive definite, then, because of exercise, the sign of the determinant of the Gram matrix equals $(-1{)}^0=1$, and is positive. Since the restriction of the form to the linear subspaces $U_i=⟨v_1,\dots ,v_i⟩$ is also positive definite, the determinants of all relevant submatrices are positive.
If, the other way round, all determinants are positive, then fact, implies that the bilinear form is positive definite.

(2) follows from (1) by considering the negative bilinear form, that is, $-⟨-,-⟩$.

To fact