Part CCSIR NET December 2024orthogonal-pair-does-not-make-an-orthonormal-basis

Orthogonal pair does not make an orthonormal basis

Let V be the $R -$ vector space of real valued continuous functions on the interval $[0, π]$ with the inner product given by ⟨f, g⟩ $= \int_{0}^{π} f (x) g (x)$ dx. Let S = { $sin (x), cos (x), sin^{2} (x), cos^{2} (x)$ } and W be the subspace of V generated by S. Which of the following statements are true?

A.S is a basis of W.✓
B.S is an orthonormal basis of W.
C.There exist f, g ∈ S such that ⟨f, g⟩ = 0.✓
D.S contains an orthonormal basis of W.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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Orthogonal pair does not make an orthonormal basis

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