A set \(V\) of complex numbers forms a two-dimensional real vector space under the usual addition of complex numbers and multiplication by real numbers. Let \(T : V \to V\) be a linear transformation defined as \(T(z) = \bar{z}\). Eigenvalues of \(T\) are:
\(1, -1\)
Choose the real basis \(\{1, i\}\) of \(V\). Then \(T(1) = \bar{1} = 1\) and \(T(i) = \bar{i} = -i\).
The matrix of \(T\) in this basis is \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\), which is diagonal with entries \(1\) and \(-1\).
Therefore the eigenvalues are \(1\) and \(-1\), corresponding respectively to the real axis (fixed) and the imaginary axis (reflected).
Hence, the eigenvalues of \(T\) are \(1, -1\).
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