Mathematics · Linear Algebra
Let B be a 4 × 4 positive-definite real symmetric matrix which is not the identity matrix. Consider the inner product on &mathbb;R4 given by ⟨v, w⟩ = vTBw, where vT denotes the transpose of v. Which of the following statements is FALSE?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1If W ⊆ &mathbb;R4 is an eigenspace of B, then W has an orthonormal basis.
- 2For every subspace W ⊆ &mathbb;R4, W + W⊥ = &mathbb;R4, where W⊥ = { v ∈ &mathbb;R4 | ⟨v, w⟩ = 0 for all w ∈ W }.
- 3If v is an eigenvector of B and w ≠ 0 is such that ⟨w, v⟩ = 0, then w is an eigenvector of B.✓
- 4If v and w are eigenvectors of B for distinct eigenvalues, then ⟨v, w⟩ = 0.
Correct answer
If v is an eigenvector of B and w ≠ 0 is such that ⟨w, v⟩ = 0, then w is an eigenvector of B.
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