Teaching CSIR NET Mock Test Series Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Let \(\rm A=\left(\begin{array}{ll} 1 & 2 \\ 4 & 3 \end{array}\right)\) and ϕ : ℝ2 → ℝ2 → ℝ be the bilinear map defined by
ϕ(v, w) = vT Aw. 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 exist a 2 × 2 symmetric matrix B such that ϕ(v, v) = vTBv for all v ∈ ℝ2.
4
The map ψ : ℝ4 → ℝ defined by ψ ((v1, v2, v3, v4)t) = ϕ((v1, v2,)t, (v3, v4)t) is linear.