Consider the hypotheses for the general linear model, which are of the form

\[H_{0}: \mathbf{T} \beta=\mathbf{c}, \quad H_{1}: \mathbf{T} \beta eq \mathbf{c}\]

where $\mathbf{T}$ is a $q \times p$ matrix of rank $q$. Derive the appropriate $F$ statistic under both the null and alternative hypothesis.