Part BCSIR NET December 2025orthogonal-to-one-eigenvector-is-a-3-dimensional-space-not-automatically-another-eigenspace

Orthogonal to one eigenvector is a 3 dimensional space not automatically another eigenspace

Let B be a 4×4 positive-definite real symmetric matrix which is not the identity matrix. Consider the inner product on $\mathbb{R}^{4}$ given by ⟨v,w⟩ = vᵀBw, where vᵀ denotes the transpose of v. Which of the following statements is FALSE?

A.If v and w are eigenvectors of B for distinct eigenvalues, then ⟨v,w⟩ = 0.
B.If v is an eigenvector of B and w ≠ 0 is such that ⟨w,v⟩ = 0, then w is an eigenvector of B.✓
C.For every subspace $W \subseteq \mathbb{R}^{4}, W + W\perp = \mathbb{R}^{4}$ , where W⊥ = { $v \in \mathbb{R}^{4}$ ∣ ⟨v,w⟩ = 0 for all w ∈ W}.
D.If $W \subseteq \mathbb{R}^{4}$ is an eigenspace of B, then W has an orthonormal basis.

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 to one eigenvector is a 3 dimensional space not automatically another eigenspace

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