= Solution
For a <group representation> $D:G\to\operatorname{GL}(V)$, a <faithful representation> is injective, equivalently $D(g)=I$ only for $g=e$. Thus no distinct group elements are represented by the same linear map.
Two <equivalent representations> are related by one fixed invertible change of basis: $E(g)=S^{-1}D(g)S$ for every $g$. The same $S$ must work simultaneously for all group elements.
An <irreducible representation> acts on a nonzero vector space and has no invariant subspace other than $\{0\}$ and the whole space. An <invariant subspace> is one carried into itself by every representing matrix.
The <character of a representation> is
$$
\boxed{\chi_D(g)=\operatorname{tr}D(g).}
$$
The <cyclic property of the trace> makes it invariant under equivalence and constant on conjugacy classes. Its value at the identity is the dimension of the representation.
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