Let V be the vector space of complex numbers over the field r. Define T:V-->V by T(z) = Z', where Z' is the complex conjugate of z.
Prove that T is linear, and compute [T]_beta, where beta={1,i}
$\displaystyle T(x+y) = \overline{x+y} = \overline{x}+\overline{y} = T(x) + T(y)$
$\displaystyle T(rx) = \overline{rx} = r\overline{x} = rT(x)\,\,\, [\because r \in R, \overline{r} = r]$
Clearly the matrix of the linear transformation $\displaystyle T \equiv \begin{pmatrix} 1 & 0 \\0 & -1\end{pmatrix}$