GATE 2026 MA – Question 37

Linear Algebra · Inner Products and Quadratic Forms · 2 marks · Multiple choice

Define $\langle x,y\rangle_P=x^TPy$ on ℝ³ with P = diag(1,9,16). For V = span{(1,0,0)ᵀ,(1,1,1)ᵀ}, choose an orthonormal basis.

$\{(1,0,0)^T,(0,1,1)^T\}$
$\{(0,1,0)^T,(0,1,1)^T\}$
$\{(1,0,0)^T,(0,1/5,1/5)^T\}$
$\{(1,0,0)^T,(0,1/\sqrt2,1/\sqrt2)^T\}$

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Correct answer: (C) $\{(1,0,0)^T,(0,1/5,1/5)^T\}$

Explanation

First vector e₁ has norm 1. Subtract its projection from (1,1,1) to get (0,1,1), whose P-norm is √(9+16) = 5. Dividing by 5 gives (0,1/5,1/5). It is P-orthogonal to e₁, yielding **C**.

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