Part BCSIR NET June 2023positive-definiteness

Positive definiteness

Let $x = (x_{1}, \dots, x_{n})$ and $y = (y_{1}, \dots, y_{n})$ denote vectors in $\mathbb{R}^{n}$ for a fixed n ≥ 2. Which of the following defines an inner product on $\mathbb{R}^{n}$ ?

A.⟨x, y⟩ $= \sum_{i,j=1}^n$ $x_{i}y_{j}$
B.⟨x, y⟩ $= \sum_{i,j=1}^n$ $(x_{i}^{2} + y_{j}^{2})$
C.⟨x, y⟩ $= \sum_{j=1}^n$ $j^{3} x_{j}y_{j}$ ✓ Correct
D.⟨x, y⟩ $= \sum_{j=1}^n$ $x_{j} y_{n}$ ₋ $_{j}$ ₊ $_{1}$

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

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Positive definiteness

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