Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Consider the quadratic form Q(v) = vtAv, where \(A=\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array}\right]\) v = (x, y, z, w)
Then
1
Q has rank 3.
2
xy + z2 = Q(Pv) for some invertible 4 x 4 real matrix P
3
xy + y2 + z2 = Q(Pv) for some invertible 4 x 4 real matrix P
4
x2 + y2 – zw = Q(Pv) for some invertible 4 x 4 real matrix P.
5
Question Not Attempted