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