Teaching CSIR NET Mock Test Series Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Let \(\rm A = \begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&1\\\ 0&0&1&1\end{pmatrix}\) and consider the symmetric bilinear form on R4 given by (v, w) = vt Aw, for v, w ∈ ℝ4. Which of the following statements is true?
1
A is invertible
2
There exist non-zero vectors v, w such that 〈v, w〉 = 0
3
〈u, v〉 ≠ 〈u, w〉 for all non-zero vectors u, v, w with v ≠ w
4
Every eigenvalue of A2 is positive