Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Let \(A=\left[\begin{array}{ll}1 & 2 \\ 4 & 3\end{array}\right] \in M_{2}(\mathbb{R})\) and ϕ : ℝ2 × ℝ2 → ℝ be the bilinear map defined by ϕ(v, w) = vTAw. Choose the correct statement from below:
1
ϕ(v, w) = ϕ(w, v) for all v, w ∈ ℝ2
2
there exists nonzero v ∈ ℝ2 such that ϕ(v, w) = 0 for all w ∈ ℝ2
3
there exists a 2 × 2 symmetric matrix B such that ϕ(v, v) = vTBv for all v ∈ ℝ2
4
the map ψ : ℝ4 → ℝ defined by
\(\psi\left(\left[\begin{array}{c} V_1 \\ V_2 \\ W_1 \\ W_2 \end{array}\right]\right)=\phi\left(\left[\begin{array}{c} V_1 \\ V_2 \end{array}\right],\left[\begin{array}{l} W_1 \\ W_2 \end{array}\right]\right)\) is linear
5
Question Not Attempted