3.9.3 Introducing Further Explanatory Variables, 3.9.4 Introducing Linear Restrictions

If $\mathbf{P}$ projects onto $\mathcal{C}(\mathbf{X})$, show that $\mathbf{Z}^{T}\left(\mathbf{I}_{n}-\mathbf{P}\right) \mathbf{Z}$ is nonsingular.

Prove that if $\mathbf{X}_{1}$ is $n \times r$ of rank $r$ and consists of a set of $r$ linearly independent columns of $\mathbf{X}$, then $\mathbf{X} = \mathbf{X}_{1} \mathbf{L}$, where $\mathbf{L}$ is $r \times p$ of rank $r$.

Prove that $\mathbf{B}$ has full column rank.

If $\mathbf{X}$ has full rank and $\widehat{\mathbf{\theta }}_{\omega } = \mathbf{X} \widehat{\mathbf{\beta }}_{H}$, show that

Show how to modify the theory above to take care of the case when the restrictions are $\mathbf{A} \mathbf{\beta } = \mathrm{c}(\mathrm{c} \neq 0)$.