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

Projection needs only the subspace finite

Consider the real vector space V=R[x]V = \mathbb{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?

  1. A.There exists a polynomial p(x) ∈ W such that x4−p(x)∈W⊥x^{4} - p(x) \in W^\perp✓ Correct
  2. B.W^⊥ = {0}
  3. C.W and W^⊥ have the same dimension over R\mathbb{R}
  4. D.W^⊥ is an infinite dimensional vector space over R\mathbb{R}✓ Correct

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, and why this one tests finite-dimensional intuition.

See pricing

50 are analysed free — try those first.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

Drill statements like this

Related counterexample: A real matrix with all real eigenvalues is orthogonally diagonalisable

More on this topic

The chapter behind this: Inner products, orthogonality and the spectral theorem — free to read

From Inner Product Spaces and Forms › Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

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