Part CCSIR NET December 2024positive-definite-says-nothing-about-this-basis

Positive definite says nothing about this basis

Let V be a two-dimensional real vector space with a basis { $v_{1}, v_{2}$ }. Let $f : V \times V \to R$ be a symmetric bilinear form. Let $r = f (v_{1}, v_{1}), s = f (v_{1}, v_{2}), t = f (v_{2}, v_{2})$ . Which of the following statements are necessarily true?

A. $f (v_{2}, v_{1}) = - s$ .
B.If r = s = t, then either f(v, v) ≥ 0 for all v ∈ V or f(v, v) ≤ 0 for all v ∈ V.✓
C.If f is positive definite, then r ≠ s, s ≠ t and r ≠ t.
D.If f is positive definite, then $f (v_{1}, v_{2}) = 0$ .

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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50 are analysed free — try those first.

Positive definite says nothing about this basis

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