GATE 2021 MA – Question 53
For an inner product ⟨·,·⟩ on ℝⁿ, P:|⟨u,v⟩|≤(⟨u,u⟩+⟨v,v⟩)/2 for all u,v. Q:if ⟨u,v⟩=⟨2u,−v⟩ for every v, then u=0. Choose truth values.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (A) Both P and Q true
Explanation
Cauchy–Schwarz and 2ab≤a²+b² prove P. In Q, bilinearity gives ⟨u,v⟩=−2⟨u,v⟩, hence ⟨u,v⟩=0 for every v. Take v=u and use positive definiteness to obtain u=0. Both true, **A**.