Discussion
Let V, W be two complex vector spaces and \(T \in L(V, W)\). If T has matrix representation :
\[ \begin{bmatrix} 2 & 1 + i & 3 \\ 4 + i & 1 - i & i \end{bmatrix} \]
Which of the following matrices represents the adjoint map \(T^{*}\) ?
Answer: C
\(T^{*}\) is represented by the conjugate transpose, \(\begin{pmatrix} 2 & 4 - i \\ 1 - i & 1 + i \\ 3 & -i \end{pmatrix}\).
With respect to orthonormal bases, the matrix of the adjoint is the conjugate transpose \(A^{*} = \bar{A}^{T}\). Transpose the \(2 \times 3\) matrix (rows become columns) and conjugate every entry: \(1 + i \to 1 - i\), \(4 + i \to 4 - i\), \(1 - i \to 1 + i\), \(i \to -i\).
Option B is the transpose without conjugation, option A conjugates without transposing (and has the wrong shape for a map from W to V), and option D is the original matrix.
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