Part CCSIR NET June 2024projection-needs-only-the-subspace-finite

Projection needs only the subspace finite

Consider the real vector space $V = R [x]$ equipped with an inner product. Let W be the subspace of V consisting of polynomials of degree at most 2. Let W^⊥ denote the orthogonal complement of W in V. Which of the following statements are true?

A.There exists a polynomial p(x) ∈ W such that $x^{4} - p(x) \in W^$ ⊥✓
B.W^⊥ = {0}
C.W and W^⊥ have the same dimension over $R$
D.W^⊥ is an infinite dimensional vector space over $R$ ✓

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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Projection needs only the subspace finite

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