= Solution
A \b[<symmetry-closed orbital basis>], often called a complete basis in this context, is a linearly independent set whose span is carried into itself by every group operation. No transformed basis function requires a function outside that span. This is completeness for the chosen <group representation>, not a claim that a small atomic-orbital set spans the full electronic Hilbert space.
A \b[representation matrix] $D(g)$ specifies the linear transformation produced by a <molecular symmetry operation> $g$. With the column convention,
$$
\hat U(g)\phi_j=\sum_i\phi_iD_{ij}(g),\qquad
D(gh)=D(g)D(h).
$$
Its \b[<character of a representation>] is the <matrix trace>,
$$
\boxed{\chi(g)=\operatorname{tr}D(g)=\sum_iD_{ii}(g).}
$$
The character is unchanged by a simultaneous change of basis and is constant on a <conjugacy class>, which is why character tables have one column per class.
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