Part BCSIR NET June 2024bilinear-form-is-not-an-inner-product

Bilinear form is not an inner product

Let A = ((0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 1, 1), (0, 0, 1, 1)), and consider the symmetric bilinear form on $R^{4}$ given by ⟨v, w⟩ = vᵗAw, for $v, w \in R^{4}$ . Which of the following statements is true?

A.A is invertible
B.There exist non-zero vectors v, w such that ⟨v, w⟩ = 0✓
C.⟨u, v⟩ ≠ ⟨u, w⟩ for all non-zero vectors u, v, w with v ≠ w
D.Every eigenvalue of $A^{2}$ is positive

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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Bilinear form is not an inner product

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