Teaching CSIR NET Mock Test Series Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Let q1(x1, x2) and q2(y1, y2) be real quadratic forms such that there exist (u1, u2), (v1, v2) ∈ ℝ2 such that q1 (u1, u2) = 1 = q2(v1, v2). Define q(x1, x2, y1, y2) = q1 (x1, x2) - q2(y1, y2). Which of the following statements are necessarily true?
1
q is a quadratic form in x1, x2, y1, y2
2
There exists (t1, t2) ∈ ℝ2 such that q1 (t1, t2) = 5
3
There does not exist (s1, s2) ∈ ℝ2 such that q2(s1, s2) = -5
4
Given α ∈ ℝ, there exists a vector ω ∈ ℝ4 such that q(ω) = α