= Solution
A <group representation> is a homomorphism $D:G\to\operatorname{GL}(V)$ into invertible linear maps. It is a <faithful representation> when it is injective, equivalently when its kernel contains only the identity. A faithful real two-dimensional representation of the <cyclic group of order four> is
$$
\boxed{D(1)=I,\quad D(i)=J=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\quad D(-1)=-I,\quad D(-i)=-J.}
$$
These are rotations of the plane through $0,\pi/2,\pi,3\pi/2$. Since $J^2=-I$ and $J^4=I$, their multiplication agrees with the group, and all four matrices are distinct.
For any invertible $S$, set $\widetilde D(g)=SD(g)S^{-1}$. Then
$$
\widetilde D(g)\widetilde D(h)=SD(g)S^{-1}SD(h)S^{-1}=SD(gh)S^{-1}=\widetilde D(gh),
$$
so this also is a <group representation>. The <character of a representation> is $\chi_D(g)=\operatorname{tr}D(g)$. For the displayed quarter-turn representation, its values on $1,i,-1,-i$ are $2,0,-2,0$.
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