Discussion
Which of the functions defines an inner product on \(\mathbf{C}^{2}\) ? Let \(x = [x_{1}, x_{2}]^{t}\), \(y = [y_{1}, y_{2}]^{t}\) ?
(i) \((x, y) = x_{1}\bar{y}_{2}\)
(ii) \((x, y) = x_{1}\bar{y}_{1} + x_{2}\bar{y}_{2}\)
(iii) \((x, y) = x_{1}y_{1} + x_{2}y_{2}\)
Answer: C
Only (ii) defines an inner product on \(\mathbf{C}^{2}\).
(ii) is the standard Hermitian inner product: it is linear in \(x\), conjugate symmetric, and \((x, x) = |x_{1}|^{2} + |x_{2}|^{2} \gt 0\) for \(x \ne 0\).
(i) fails positivity: \(x = (1, 0)\) gives \((x, x) = 1 \cdot \bar{0} = 0\) with \(x \ne 0\). (iii) fails over \(\mathbf{C}\): \(x = (i, 0)\) gives \((x, x) = i^{2} = -1\), and it is not conjugate symmetric. So options naming (i) or (iii) are wrong.
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