GATE 2025 ST – Question 13

Matrix Theory · Inner Products and Quadratic Forms · 1 mark · Multiple choice

Let S={(x,y,z)∈ℝ³∖{0}:z=−(x+y)}. Define S⊥={v∈ℝ³:v·s=0 for every s∈S}. Which holds?

S⊥ is not a subspace.
S⊥={0}.
dim S⊥=1
dim S⊥=2

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) dim S⊥=1

Explanation

Removing zero from the plane does not alter its orthogonality conditions or its span. The plane has normal (1,1,1), so S⊥=span(1,1,1), a one-dimensional subspace. **C**.

More Inner Products and Quadratic Forms questions

GATE 2024 ST Q15 – In standard ℝ², u=(a,b)ᵀ has inner products 2 with (1,2)ᵀ and −1 with (4,−2)ᵀ. Which…
GATE 2023 ST Q36 – For a real n×n matrix A: I:if A is symmetric,there exists c≥0 for which A+cI is symmetric…
GATE 2023 ST Q59 – In ℝ³ with standard inner product,the orthonormal basis is…
GATE 2022 ST Q23 – A unit vector u∈ℝ⁴ is orthogonal to S=x:x₁+x₂+x₃+x₄=0. Let v=(1,−2,1,1)ᵀ. If u and v−αu…
GATE 2022 ST Q36 – A symmetric 3×3 matrix M has eigenvalues −1,1,2 and corresponding unit eigenvectors u,v,w.…
GATE 2022 ST Q47 – M is symmetric 3×3 with eigenvalues 1,2,3. N has real eigenvalues and MN+NᵀM=3I. Which…

All Inner Products and Quadratic Forms questions →