Teaching CSIR NET Mock Test Series Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Consider the quadratic form Q(x, y, z) = x2 + xy + y2 + xz + yz + z2. Which of the following statements are true?
1
There exists a non-zero u ∈ ℚ3 such that Q(u) = 0.
2
There exists a non-zero u ∈ ℝ3 such that Q(u) = 0.
3
There exist a non-zero u ∈ ℂ3 such that Q(u) = 0.
4
The real symmetric 3 × 3 matrix A which satisfies Q(x, y, z) = [x y z] A\(\)[x,y,z]t for all x, y, z ∈ ℝ is invertible.