I don't really know what $[\phi_i(v_j)]$ really is. As far as I understand, $\phi_i$ is a linear transformation - a matrix; and $v_j$ is the column vector it eats. So $[\phi_i(v_j)]$ spit out column vectors, rather than $k \times k$ real matrix as the problem stated.
Again, thanks for your help~
The dual space $V^\star$ contains linear maps into the reals $\mathbb R$ and so for each $i,j$ we obtain a real number. Them thinking of $i,j$ being indices in a table, this defines a matrix.