  For $n\geq 2$, we determine the Dirichlet spectrum in $\Rn$ with respect to a
linear form and the maximum norm as the entire interval $[0,1]$. This natural
result improves on recent work of Beresnevich, Guan, Marnat, Ram\'irez and
Velani, and complements a subsequent paper by the author where the analogous
result was proved for simultaneous approximation. Various generalisations that
can be obtained by similar methods as in the latter paper are indicated. We
believe that our results are an important step towards resolving the very open
analogous problem for a general system of linear forms.
