GATE Linear Algebra: Inner Products and Quadratic Forms – Previous Year Questions
8 GATE previous year questions on Inner Products and Quadratic Forms (Linear Algebra, Mathematics) with answers and explanations, from every paper.
- GATE 2021 MA Q53 – 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.
- GATE 2023 MA Q34 – In ℝ⁴ with the standard inner product, M=x:x₁=x₃. Find dim M⊥.
- GATE 2024 MA Q42 – Five distinct unit vectors v₁,…,v₅ in ℝ⁷ have nonpositive integer pairwise inner products for unequal indices. N counts ordered pairs (r,s), including…
- GATE 2025 MA Q61 – Under ⟨f,g⟩=∫₋₁¹f(x)g(x)dx, polynomial p(x)=α+βx²−30x⁴ is orthogonal to every polynomial of degree at most three. Find α+5β (integer).
- GATE 2025 MA Q62 – M is the symmetric matrix of Q=2x₁²+2x₂²+3x₃²+4x₁x₂+2x₁x₃+2x₂x₃. For any nonzero Y∈ℝ³, a n=YT(M+I)n+1Y/[YT(M+I)nY]. Find lim a n (integer).
- GATE 2026 MA Q21 – P is real symmetric positive definite, 3×3. Which statements about QTPQ are true?
- GATE 2026 MA Q37 – Define x,y P=xTPy on ℝ³ with P = diag(1,9,16). For V = span(1,0,0)ᵀ,(1,1,1)ᵀ, choose an orthonormal basis.
- GATE 2026 MA Q57 – Maximize z=xTpmatrix2&0&0\\0&2&1\\0&1&2pmatrixx over real vectors with xᵀx=1. Find the maximum (integer).