GATE 2023 MA – Question 34

Linear Algebra · Inner Products and Quadratic Forms · 1 mark · Numerical answer

In ℝ⁴ with the standard inner product, M={x:x₁=x₃}. Find dim M⊥.

Show answer and explanation

Correct answer: 1 to 1

Explanation

M is the kernel of the nonzero functional x₁−x₃, hence has dimension 3. Its orthogonal complement is span{(1,0,−1,0)}, of dimension **1**.

GATE 2021 MA Q53 – For an inner product ⟨·,·⟩ on ℝⁿ, P: ⟨u,v⟩ ≤(⟨u,u⟩+⟨v,v⟩)/2 for all u,v. Q:if…
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…
GATE 2026 MA Q57 – Maximize z=xTpmatrix2&0&0\\0&2&1\\0&1&2pmatrixx over real vectors with xᵀx=1. Find the…
GATE 2025 MA Q61 – Under ⟨f,g⟩=∫₋₁¹f(x)g(x)dx, polynomial p(x)=α+βx²−30x⁴ is orthogonal to every polynomial…
GATE 2025 MA Q62 – M is the symmetric matrix of Q=2x₁²+2x₂²+3x₃²+4x₁x₂+2x₁x₃+2x₂x₃. For any nonzero Y∈ℝ³, a…